analytic and differential geometry mikhail g. katz∗ - cs@biu

Analytic and Differential geometry

Mikhail G. Katz∗

Department of Mathematics, Bar Ilan University, Ra-

mat Gan 52900 Israel

Email address : [email protected]

∗Supported by the Israel Science Foundation (grants no. 620/00-10.0and 84/03).

Abstract. We start with analytic geometry and the theory ofconic sections. Then we treat the classical topics in differentialgeometry such as the geodesic equation and Gaussian curvature.Then we prove Gauss’s theorema egregium and introduce the ab-stract viewpoint of modern differential geometry.

Contents

Chapter 1. Analytic geometry 91.1. Circle, sphere, great circle distance 91.2. Linear algebra, index notation 111.3. Einstein summation convention 121.4. Symmetric matrices, quadratic forms, polarisation 121.5. Matrix as a linear map 131.6. Symmetrisation and antisymmetrisation 141.7. Matrix multiplication in index notation 151.8. Two types of indices: summation index and free index 161.9. Kronecker delta and the inverse matrix 161.10. Vector product 171.11. Eigenvalues, symmetry 181.12. Euclidean inner product 19

Chapter 2. Eigenvalues of symmetric matrices, conic sections 212.1. Finding an eigenvector of a symmetric matrix 212.2. Trace of product of matrices in index notation 232.3. Inner product spaces and self-adjoint endomorphisms 242.4. Orthogonal diagonalisation of symmetric matrices 242.5. Classification of conic sections: diagonalisation 262.6. Classification of conics: trichotomy, nondegeneracy 282.7. Characterisation of parabolas 30

Chapter 3. Quadric surfaces, Hessian, representation of curves 333.1. Summary: classification of quadratic curves 333.2. Quadric surfaces 333.3. Case of eigenvalues (+ + +) or (−−−), ellipsoid 343.4. Determination of type of quadric surface: explicit example 353.5. Case of eigenvalues (+ +−) or (+−−), hyperboloid 363.6. Case rank(S) = 2; paraboloid, hyperbolic paraboloid 373.7. Cylinder 393.8. Exercise on index notation 403.9. Gradient and Hessian 403.10. Minima, maxima, saddle points 41

3

4 CONTENTS

3.11. Parametric representation of a curve 423.12. Implicit representation of a curve 433.13. Implicit function theorem 43

Chapter 4. Curvature, Laplacian, Bateman–Reiss 474.1. Unit speed parametrisation 474.2. Geodesic curvature of a curve 474.3. Tangent and normal vectors 484.4. Osculating circle of a curve 504.5. Center of curvature, radius of curvature 514.6. Flat Laplacian 524.7. Bateman–Reiss operator 524.8. Geodesic curvature for an implicit curve 534.9. Curvature of circle via DB 544.10. Curvature of parabola via DB 544.11. Curvature of hyperbola via DB 554.12. Maximal curvature of a logarithmic curve 564.13. Exploiting the defining equation in studying curvature 564.14. Curvature of graph of function 58

Chapter 5. Surfaces and their curvature 595.1. Existence of arclength 595.2. Surfaces; Arnold’s observation on folding a page 605.3. Regular surface; Jacobian 615.4. Coefficients of first fundamental form of a surface 625.5. Metric coefficients in spherical coordinates 625.6. Tangent plane; Gram matrix 635.7. First fundamental form Ip of M 645.8. The form Ip of plane and cylinder 645.9. Surfaces of revolution 655.10. From tractrix to pseudosphere 665.11. Chain rule in two variables 675.12. Measuring length of curves on surfaces 67

Chapter 6. Gamma symbols of a surface 696.1. Normal vector to a surface 696.2. Γ symbols of a surface 706.3. Basic properties of the Γ symbols 716.4. Derivatives of the metric coefficients 726.5. Intrinsic nature of the Γ symbols 736.6. Gamma symbols for a surface of revolution 736.7. Spherical coordinates 746.8. Great circles 75

CONTENTS 5

6.9. Position vector orthogonal to tangent vector of S2 77

Chapter 7. Clairaut’s relation, geodesic equation 797.1. Clairaut’s relation: bridge bt. geometry and analysis 797.2. Spherical sine law 797.3. Longitudes; length-minimizing property 807.4. Preliminaries to proof of Clairaut’s relation 827.5. Proof of Clairaut’s relation 837.6. Differential equation of great circle 837.7. Metrics conformal to the flat metric and their Γkij 857.8. Gravitation, ants, and geodesics on a surface 877.9. Geodesic equation on a surface 887.10. Equivalence of definitions 89

Chapter 8. Geodesics on surface of revolution, Weingarten map 918.1. Geodesics on a surface of revolution 918.2. Integrating geodesic equation on surface of rev 938.3. Area integration in polar coordinates 958.4. Integration in cylindrical coordinates in R3 968.5. Integration in spherical coordinates 968.6. Measuring area on surfaces 978.7. Directional derivative as derivative along path 988.8. Extending v.f. along surface to an open set in R3 998.9. Differentiating normal vectors n and N 1008.10. Hessian of a function at a critical point 1018.11. Definition of the Weingarten map 102

Chapter 9. Gaussian curvature; second fundamental form 1039.1. Properties of Weingarten map 1039.2. Self-adjointness of Weingarten map 1049.3. Relation to the Hessian of a function 1059.4. Weingarten map of sphere 1069.5. Weingarten map of cylinder 1069.6. Coefficients Lij of Weingarten map 1079.7. Gaussian curvature 1089.8. Second fundamental form 1099.9. Geodesics and second fundamental form 1109.10. Three formulas for Gaussian curvature 1119.11. Principal curvatures of surfaces 112

Chapter 10. Minimal surfaces; Theorema egregium 11710.1. Mean curvature, minimal surfaces 11710.2. Scherk surface 117

6 CONTENTS

10.3. Minimal surfaces in isothermal coordinates 11910.4. Minimality and harmonic functions 12110.5. Intro to theorema egregium; Riemann’s formula 12310.6. An identity involving the Γs and the Ls 12610.7. The theorema egregium of Gauss 12710.8. Signed curvature as θ′(s) 128

Chapter 11. Signed curvature of curves; total curvature 13111.1. Signed curvature with respect to arbitrary parameter 13111.2. Jordan curves; global geometry; Gauss map 13211.3. Convex Jordan curves; orientation 13311.4. Properties of Gauss map 13411.5. Total curvature of a convex Jordan curve 13511.6. Rotation index of a closed curve in the plane 13711.7. Total signed curvature of Jordan curve 13911.8. Gauss–Bonnet theorem for a plane domain 13911.9. Gaussian curvature as product of signed curvatures 14011.10. Connection to the Hessian 14111.11. Mean versus Gaussian curvature 141

Chapter 12. Laplace–Beltrami, Gauss–Bonnet 14512.1. The Laplace–Beltrami operator 14512.2. Laplace-Beltrami formula for Gaussian curvature 14512.3. Hyperbolic metric in the upper half-plane 14712.4. Area elements of the surface, orientability 14812.5. Gauss map for surfaces in R3 14912.6. Sphere S2 parametrized by Gauss map of M 15012.7. Comparison of two parametrisations 15112.8. Gauss–Bonnet theorem for surfaces with K > 0 15212.9. Proof of Gauss–Bonnet Theorem 15312.10. Gauss–Bonnet with boundary 15412.11. Euler characteristic and Gauss–Bonnet 154

Chapter 13. Rotation index, isothermisation, pseudosphere 15713.1. Rotation index of closed curves and total curvature 15713.2. Surfaces of revolution and isothermalisation 15713.3. Gaussian Curvature of pseudosphere 15913.4. Transition from classical to modern diff geom 16013.5. Duality in linear algebra; 1-forms 16013.6. Dual vector space 16113.7. Derivations 16213.8. Characterisation of derivations 16313.9. Tangent space and cotangent space 164

CONTENTS 7

13.10. Constructing bilinear forms out of 1-forms 16513.11. First fundamental form 16613.12. Dual bases in differential geometry 16613.13. More on dual bases 167

Chapter 14. Lattices and tori 16914.1. Circle via the exponential map 16914.2. Lattice, fundamental domain 17014.3. Fundamental domain 17014.4. Lattices in the plane 17114.5. Successive minima of a lattice 17214.6. Gram matrix 17314.7. Sphere and torus as topological surfaces 17314.8. Standard fundamental domain 17414.9. Conformal parameter τ of a lattice 17514.10. Conformal parameter τ of tori of revolution 17714.11. θ-loops and φ-loops on tori of revolution 17814.12. Tori generated by round circles 17814.13. Conformal parameter of tori of revolution, residues 179

Chapter 15. A hyperreal view 18115.1. Successive extensions N, Z, Q, R, ∗R 18115.2. Motivating discussion for infinitesimals 18215.3. Introduction to the transfer principle 18315.4. Transfer principle 18615.5. Orders of magnitude 18715.6. Standard part principle 18915.7. Differentiation 189

Chapter 16. Global and systolic geometry 19316.1. Definition of systole 19316.2. Loewner’s torus inequality 19716.3. Loewner’s inequality with remainder term 19816.4. Global geometry of surfaces 19816.5. Definition of manifold 19916.6. Sphere as a manifold 20016.7. Dual bases 20016.8. Jacobian matrix 20116.9. Area of a surface, independence of partition 20216.10. Conformal equivalence 20316.11. Geodesic equation 20416.12. Closed geodesic 20516.13. Existence of closed geodesic 206

8 CONTENTS

16.14. Surfaces of constant curvature 20616.15. Real projective plane 20616.16. Simple loops for surfaces of positive curvature 20716.17. Successive minima 20816.18. Flat surfaces 20916.19. Hyperbolic surfaces 20916.20. Hyperbolic plane 210

Chapter 17. Elements of the topology of surfaces 21317.1. Loops, simply connected spaces 21317.2. Orientation on loops and surfaces 21317.3. Cycles and boundaries 21417.4. First singular homology group 21517.5. Stable norm in 1-dimensional homology 21617.6. The degree of a map 21717.7. Degree of normal map of an embedded surface 21817.8. Euler characteristic of an orientable surface 21817.9. Gauss–Bonnet theorem 21917.10. Change of metric exploiting Gaussian curvature 21917.11. Gauss map 22017.12. An identity 22117.13. Stable systole 22217.14. Free loops, based loops, and fundamental group 22217.15. Fundamental groups of surfaces 223

Chapter 18. Pu’s inequality 225

Chapter 19. Approach To Loewner via energy-area identity 22719.1. An integral-geometric identity 22719.2. Two proofs of the Loewner inequality 22819.3. Remark on capacities 23019.4. A table of optimal systolic ratios of surfaces 230

Chapter 20. A primer on surfaces 23320.1. Hyperelliptic involution 23320.2. Hyperelliptic surfaces 23520.3. Ovalless surfaces 23520.4. Katok’s entropy inequality 237

Bibliography 241

CHAPTER 1

Analytic geometry

(1) At http://u.math.biu.ac.il/~katzmik/88-201.html youwill find the course site.

(2) There you will find choveret in English as well as tirgul notesby Atia in Hebrew.

(3) The final exam is 90% of the grade, bochan 5%, and targil 5%.(4) The homework assignments can be found at the course site.(5) Feel free to ask questions via email: [email protected]

After dealing with classical geometric preliminaries including thetheorema egregium of Gauss, we present new geometric inequalities onRiemann surfaces, as well as their higher dimensional generalisations.

We will first review some familiar objects from classical geometryand try to point out the connection with important themes in modernmathematics.

1.1. Circle, sphere, great circle distance

Definition 1.1.1. The unit circle S1 in the plane is the locus ofthe equation

x2 + y2 = 1

in the (x, y)-plane.

The circle solves the isoperimetic problem in the plane. Namely,consider simple (non-self-intersecting) closed curves of equal perimeter,for instance a polygon.1 2 3

1metzula2Among all such curves, the circle is the curve that encloses the largest area.

In other words, the circle satisfies the boundary case of equality in the followinginequality, known as the isoperimetric inequality. Let L be the length of the Jordancurve and A the area of the finite region bounded by the curve.

Theorem. Every Jordan curve in the plane satisfies the inequality(

L2π

)2−Aπ≥

0, with equality if and only if the curve is a round circle.3The round circle is the subject of Gromov’s filling area conjecture. The Rie-

mannian circle of length 2π is a great circle of the unit sphere, equipped withthe great-circle distance. The emphasis is on the fact that the distance is mea-sured along arcs rather than chords (straight line intervals). For all the apparent

9

http://u.math.biu.ac.il/~katzmik/88-201.html

10 1. ANALYTIC GEOMETRY

Definition 1.1.2. The 2-sphere S2 is a surface that is the collectionof unit vectors in 3-sphace:

S2 ={(x, y, z) ∈ R3 : x2 + y2 + z2 = 1

}.

Definition 1.1.3. A great circle of S2 is the intersection of thesphere with a plane passing through the origin.

Example 1.1.4. The equator is an example of a great circle.

Definition 1.1.5. Let p, q ∈ S2. The great circle distance d(p, q)on S2 is the distance measured along the arcs of great circles connectinga pair of points p, q ∈ S2:

d(p, q) = arccos(p · q),where p · q is the usual dot product in Euclidean space.

Remark 1.1.6. The distance between a pair of points p, q ∈ S2

is the length of the smaller of the two arcs of the great circle passingthrough p and q. In Theorem 7.3.2 we will prove that this arc is theminimal distance between p and q among all curves on S2 joining pand q. In other words, the geodesic arc is length-minimizing among allpaths between a pair of points on the sphere.

Remark 1.1.7. Key concepts of differential geometry that we hopeto clarify in our course are the notions of geodesic curve and curvature.

Remark 1.1.8. In relativity theory, one uses a framework similarto classical differential geometry, with a technical difference having todo with the basic quadratic form being used. Nonetheless, some ofthe key concepts, such as geodesic and curvature, are common to bothapproaches. In the first approximation, one can think of relativitytheory as the study of 4-manifolds with a choice of a

“light cone”4

at every point. Einstein gave a strong impetus to the development ofdifferential geometry, as a tool in studying relativity. We will system-atically use Einstein’s summation convention; see Section 1.3.

simplicity of the the Riemannian circle, it turns out that it is the subject of astill-unsolved conjecture of Gromov’s, namely the filling area conjecture. A sur-face with a single boundary circle will be called a filling of that circle. We nowconsider fillings of the Riemannian circles such that the ambient distance does notdiminish the great-circle distance (in particular, filling by the flat unit disk is notallowed). M. Gromov conjectured that Among all fillings of the Riemannian circlesby a surface, the hemisphere is the one of least area. Partial progress was obtainedin [BCIK05].

4konus ha’or

1.2. LINEAR ALGEBRA, INDEX NOTATION 11

1.2. Linear algebra, index notation

Linear algebra provides an indispensable foundation for our subject.Key concepts here are

(1) linear map, and(2) bilinear form.

It is important to develop a facility with Einstein summation conven-tion (treated in detail in Section 1.3). This notation will be exploitedthroughout the course.

Let Rn denote the Euclidean n-space. Its vectors will be denoted

v, w ∈ Rn.

Definition 1.2.1. A vector in Rn is a column vector v. To obtaina row vector we take the transpose, vt.

Definition 1.2.2. Let Mn,n(R) be the space of n by n matrices.

Let B ∈ Mn,n(R) be an n by n matrix. There are two ways ofviewing such a matrix, either as

(1) a linear map (see Remark 1.2.3) or(2) a bilinear form (see Section 1.5).

Developing suitable notation to capture this distinction helps simplifydifferential-geometric formulas down to readable size, and also to mo-tivate the important distinction between a vector and a covector.

Remark 1.2.3 (Matrix as a bilinear form B(v, w)). Consider abilinear form

B(v, w) : Rn × Rn → R, (1.2.1)

sending the pair of vectors (v, w) to the real number

vtBw. (1.2.2)

Here vt is the row vector given by transpose of v. We write

B = (bij)i=1,...,n; j=1,...,n

so that bij is an entry while (bij), with parentheses, denotes the matrixitself.

Example 1.2.4. In the 2 by 2 case, we have

v =

(v1

v2

), w =

(w1

w2

), B =

(b11 b12b21 b22

).

Then

Bw =

(b11 b12b21 b22

)(w1

w2

)=

(b11w

1 + b12w2

b21w1 + b22w

2

).


Note that the transpose vt = (v1 v2) is a row vector. We thereforecalculate the product (1.2.2) to obtain

B(v, w) = vtBw

= (v1 v2)

(b11w

1 + b12w2

b21w1 + b22w

2

)

= b11v1w1 + b12v

1w2 + b21v2w1 + b22v

2w2,

and hence vtBw =∑2

i=1

∑2j=1 bijv

iwj.

We would like to simplify this notation by suppressing the summa-tion symbols “Σ”. The details appear in Section 1.3.

1.3. Einstein summation convention

The following useful notational device was originally introduced byAlbert Einstein.

Definition 1.3.1. The rule is that whenever a product contains asymbol with a certain lower index and another symbol with the sameupper index, take summation over this repeated index, even thoughthe summation symbol Σ is not present.

Example 1.3.2. Using this notation, the bilinear form (1.2.1) de-fined by the matrix B can be written as follows:

B(v, w) = bijviwj,

with implied summation over both indices i and j.

To avoid any risk of ambiguity when using the Einstein summationconvention, we proceed as follows. When we wish to consider a specificterm such as ai times vi rather than the sum over a dummy index i,we use the underline notation as follows.

Definition 1.3.3. The expression

aivi

denotes a specific term with specific index value i (rather than sum-mation over a dummy index i).

1.4. Symmetric matrices, quadratic forms, polarisation

Definition 1.4.1. Let B be a symmetric matrix: Bt = B. The as-sociated quadratic form Q is a quadratic form associated with a bilinearform B(v, w) by the following rule:

Q(v) = B(v, v)

for all vectors v.

1.5. MATRIX AS A LINEAR MAP 13

Let (ei)i=1,...,n = (e1, . . . , en) be the standard basis of Rn. Given avector v ∈ Rn, we write it as v = viei. Each of the components vi is areal number.

Lemma 1.4.2. Given a symmetric bilinear form as above, the asso-ciated quadratic form Q satisfies

Q(v) = bijvivj

in terms of the Einstein summation convention.

Proof. To compute Q(v), we must introduce an extra index j anduse it for the second occurrence of the vector v:

Q(v) = B(v, v) = B(viei, vjej) = B(ei, ej)v

ivj = bijvivj, (1.4.1)

proving the lemma. �

Theorem 1.4.3. The polarisation formula asserts that

B(v, w) =1

4(Q(v + w)−Q(v − w)).

The polarisation formula allows one to reconstruct the symmetricbilinear form from the quadratic form.5 For an application, see Sec-tion 13.10.

1.5. Matrix as a linear map

Given a real matrix B, consider the associated map

BR : Rn → Rn, v 7→ Bv.

In order to distinguish this case from the case of the bilinear form, wewill raise the first index of the matrix coefficients. Thus, we write B as

B = (bij)i=1,...,n; j=1,...,n

where it is important to s t a g g e r the indices as follows.

Definition 1.5.1. Staggering the indices meaning that we do notplace j under i as in

bij,

but, rather, leave a blank space (in the place were j used to be), as in

bij.

5This is true whenever the characteristic of the background field is not 2. Ourbase field R has characteristic 0 and therefore the formula applies in this case.


Consider vectors v = (vj)j=1,...,n and w = (wi)i=1,...,n. Then theequation w = Bv can be written as a system of n scalar equations,

wi = bijvj for i = 1, . . . , n

using the Einstein summation convention (here the repeated index is j).

Definition 1.5.2. The formula for the trace Tr(B) = b11 + b22 +· · ·+ bnn in Einstein notation becomes

Tr(B) = bii

(here the repeated index is i).

Remark 1.5.3. If we wish to specify the i-th diagonal coefficientof our matrix B we use the underline notation

bii

as in Definition 1.3.3 (so as to avoid ambiguity).

1.6. Symmetrisation and antisymmetrisation

For the purposes of dealing with symmetrisation and antisymmetri-sation of a matrix, it is convenient to return to the framework of bilinearforms (i.e., both indices are lower indices).

Definition 1.6.1. The transpose Bt of a matrix B = (bij) is thematrix whose (i, j)-th coefficient is bji.

Geometrically the passage from B to Bt corresponds to reflectionin the main diagonal of the matrix B.

Definition 1.6.2. Let B = (bij). Its symmetric part S is by defi-nition

S =B + Bt

2=(bij + bji

2

)i=1,...,nj=1,...,n

,

while the antisymmetric (or skew-symmetric) part A is

A =B − Bt

2=(bij − bji

2

)i=1,...,nj=1,...,n

.

Theorem 1.6.3. We have B = S + A.

Another useful notation is that of symmetrisation and antisym-metrisation.

Definition 1.6.4. Symmetrisation is defined by setting

b{ij} =1

2(bij + bji) (1.6.1)

1.7. MATRIX MULTIPLICATION IN INDEX NOTATION 15

with curly braces,6 and antisymmetrisation by setting

b[ij] =1

2(bij − bji). (1.6.2)

Lemma 1.6.5. A matrix B = (bij) is symmetric if and only if forall indices i and j one has b[ij] = 0.

Proof. We have b[ij] = 12(bij − bji) = 0 since symmetry of B

means bij = bji. �

1.7. Matrix multiplication in index notation

The usual way to define matrix multiplication is as follows. Atriple of matrices A = (aij), B = (bij), and C = (cij) satisfy theproduct relation C = AB if, introducing an additional dummy index k

(cf. formula (1.4.1)), we have the relation cij =∑

k

aikbkj.

Example 1.7.1 (Skew-symmetrisation of matrix product). By com-mutativity of multiplication of real numbers, we have

aikbkj = bkjaik.

Then the coefficients c[ij] of the skew-symmetrisation of the matrix C =AB satisfy

c[ij] =∑

k

bk[jai]k.

Here by definition

bk[jai]k =1

2(bkjaik − bkiajk).

This notational device will be particularly useful in writing downthe theorema egregium of Gauss (see Section 10.7). Given below are afew examples of symmetrisation and antisymmetrisation notation:

• See Section 6.2, where we will use formulas of type

gmjΓmik + gmiΓ

mjk = 2gm{jΓ

mi}k;

• See Section 9.7 for Li[jLkℓ];

• See Section 10.6 for Γki[jΓnℓ]m.

Recall that a real matrix A naturally defines an endomorphismAR : Rn → Rn.

6Sograyim mesulsalim or metultalim


Theorem 1.7.2. If A = (aij), B = (bij), and C = (cij) then theproduct relation C = AB corresponding to the composition of linearmaps CR = AR ◦BR simplifies to the relation

cij = aikbkj for all i, j. (1.7.1)

The proof is immediate from the definition of matrix multiplication.

Remark 1.7.3. The index notation we have described reflects thefact that the natural products of matrices are the ones which corre-spond to composition of maps.

1.8. Two types of indices: summation index and free index

In expressions of type (1.7.1) it is important to distinguish betweentwo types of indices: a free index or an summation index.

Definition 1.8.1. An index appearing both as a subscript and asuperscript is called an summation index. The remaining indices arecalled free.

A summation index is often referred to as a dummy index in theliterature.

Example 1.8.2. In formula (1.7.1) the index k is a summationindex whereas indices i and j are free indices.

1.9. Kronecker delta and the inverse matrix

The Kronecker delta function δ defined as follows.

Definition 1.9.1. The expression

δij =

{1 if i = j0 if i 6= j

is called the Kronecker delta function.

Consider a matrix B = (bij).

Definition 1.9.2. The inverse matrix B−1 is the matrix

B−1 = (bij) i=1,...,nj=1,...,n

where both indices have been raised.

The identity matrix is denoted I = (δij). Then the equationB−1B =I becomes

bikbkj = δij (1.9.1)

in Einstein notation with repeated index k.

1.10. VECTOR PRODUCT 17

Remark 1.9.3. In (1.9.1) the index k is a summation index whereas iand j are free indices.

Example 1.9.4. The identity endomorphism I = (δij) by definition

satisfies AI = A = IA for all endomorphisms A = (aij), or equivalently

aijδjk = aik = δija

jk, (1.9.2)

using the Einstein summation convention.

Remark 1.9.5. In expression (1.9.2) the index j is a summationindex whereas indices i and k are free indices.

Example 1.9.6. Let δij be the Kronecker delta function on Rn,where i, j = 1, . . . , n. We view the Kronecker delta function as a lineartransformation Rn → Rn.

(1) Evaluate the expression δijδjk paying attention to which the

summation indices are;(2) Evaluate the expression δijδ

ji paying attention to which the

summation indices are.

1.10. Vector product

We briefly review the following material from linear algebra.

Definition 1.10.1. Given a pair of vectors v = viei and w = wjejin R3, their vector product is a vector v ×w ∈ R3 satisfying one of thefollowing two equivalent conditions.

(1) (algebraic) Let i = e1, j = e2, k = e3. We have v × w =

det

i j kv1 v2 v3

w1 w2 w3

, in other words,

v × w = (v2w3 − v3w2)i− (v1w3 − v3w1)j + (v1w2 − v2w1)k.

(2) (geometric) the vector v×w is perpendicular to both v and w,of length equal to the area of the parallelogram spanned bythe two vectors, and furthermore satisfying the right handrule,7 meaning that the 3 by 3 matrix formed by the threevectors v, w, and v × w has positive determinant.

Theorem 1.10.2. We have an identity

a× (b× c) = (a · c)b− (a · b)c (1.10.1)

for every triple of vectors a, b, c in R3.

7Klal yad yamin


Note that both sides of (1.10.1) vanish if a is perpendicular to both band c.

1.11. Eigenvalues, symmetry

Properly understanding surface theory and related key conceptssuch as the Weingarten map (see Section 8.11) depends on linear-algebraic background related to diagonalisation of

(1) symmetric matrices or, more generally,(2) selfadjoint endomorphisms.

The following fact was discussed in linear algebra.

Proposition 1.11.1. A real matrix may have no real eigenvectoror eigenvalue.

Example 1.11.2. The matrix of a 90 degree rotation in the plane,(0 −11 0

)

does not have a real eigenvector for obvious geometric reasons (nodirection is preserved but rather rotated).

In this section we prove the existence of a real eigenvector (andhence, a real eigenvalue) for a real symmetric matrix and a self-adjointendomorphism.

This fact has important ramifications in surface theory, since thevarious notions of curvature of a surface are defined in terms of theeigenvalues of a selfadjoint endomorphism called the Weingarten map(see Sections 8.7 and 9.11) which is a coordinate-independent formula-tion of the notion of a symmetric matrix. Let

I =

1 0

. . .0 1

be the (n, n) identity matrix. Thus

I = (δij) i=1,...,nj=1,...,n

where δij is the Kronecker delta.

Definition 1.11.3. Let B be an (n, n)-matrix. A scalar λ is calledan eigenvalue of B if

det(B − λI) = 0.

1.12. EUCLIDEAN INNER PRODUCT 19

Theorem 1.11.4. If λ ∈ R is an eigenvalue of B, then there is avector v ∈ Rn, v 6= 0, such that

Bv = λv. (1.11.1)

The proof was given in linear algebra and is not reproduced here.

Definition 1.11.5. A nonzero vector satisfying (1.11.1) is calledan eigenvector belonging to λ.

1.12. Euclidean inner product

Definition 1.12.1. The Euclidean inner product of vectors v, w ∈Rn is defined by

〈v, w〉 = v1w1 + · · ·+ vnwn =n∑

i=1

viwi.

Recall that all of our vectors are column vectors.

Lemma 1.12.2. The inner product can be expressed in terms ofmatrix multiplication as follows: 〈v, w〉 = vt w.

Recall the following theorem from basic linear algebra.

Theorem 1.12.3. The transpose has the following property:

(AB)t = BtAt.

Recall the following definition (see Section 1.6).

Definition 1.12.4. A square matrixA is called symmetric ifAt = A.

Lemma 1.12.5. Let B be a real matrix. Then the following twoconditions are equivalent:

(1) B symmetric;(2) for all v, w ∈ Rn, one has 〈Bv,w〉 = 〈v,Bw〉.Proof. We have

〈Bv,w〉 = (Bv)tw = vtBtw = 〈v,Btw〉 = 〈v,Bw〉,proving the direction (1) =⇒ (2). Conversely, if v = ei and w = ejthen 〈Bv,w〉 = bij while 〈v,Bw〉 = bji, proving the other direction(1) ⇐= (2). �

CHAPTER 2

Eigenvalues of symmetric matrices, conic sections

2.1. Finding an eigenvector of a symmetric matrix

In this section we continue with linear-algebraic preliminaries forthe theory of surfaces in Euclidean space. Eigenvalues and eigenvectorswere reviewed in Section 1.11.

Definition 2.1.1. The Hermitian product on Cn is the product

〈z, w〉 =n∑

i=1

ziwi ∀z, w ∈ Cn, (2.1.1)

where wi is the complex conjugate of wi.

Thus the Hermitian inner product on Cn is linear in one variableand skew-linear in the other.1

Theorem 2.1.2. Every real symmetric matrix possesses a real eigen-vector.

We will give two proofs of this important theorem. The first proofis simpler, more algebraic, and passes via complexification. The secondproof is more geometric.

First proof. Let n ≥ 1, and let B ∈ Mn,n(R) be an n × n realsymmetric matrix. As such, it defines a linear map

BR : Rn → Rn, v 7→ Bv

sending a vector v ∈ Rn to the vector Bv. The prove the theorem infive steps.

Step 1. Consider the field extension R → C. Via such an inclusion,we can view B as a complex matrix B ∈Mn,n(C). Then the matrix Bdefines a complex endomorphism

BC : Cn → Cn

sending a vector v ∈ Cn to the vector Bv.

1The sum appearing in (2.1.1) is our convention. Some texts adopt the alter-

native convention 〈z, w〉 =∑ni=1 z

iwi which we will not use.

21

22 2. EIGENVALUES OF SYMMETRIC MATRICES, CONIC SECTIONS

Step 2. Let 〈 , 〉 be the standard Hermitian inner product in Cn

as in (2.1.1), extending the scalar product in Rn. Since B is real, byLemma 1.12.5, we obtain for all z ∈ Cn,

〈Bz, z〉 = 〈z, Bz〉 (2.1.2)

by symmetry B = Bt.

Step 3. The characteristic polynomial pB(λ) = det(B − λI) ofthe endomorphism BC is a polynomial of positive degree n > 0. Bythe fundamental theorem of algebra, the polynomial pB possesses aroot λ0 ∈ C.

Step 4. Let z ∈ Cn be an eigenvector belonging to λ0. Then wehave Bz = λ0z. Since the Hermitian inner product is skew-linear inthe second variable, equality (2.1.2) gives

〈λ0z, z〉 = 〈z, λ0z〉 = λ0〈z, z〉.Therefore λ0〈z, z〉 = λ0〈z, z〉 so that λ0 = λ0, i.e., λ0 is a real eigen-value.

Step 5. Since λ0 is a real root of pB, there exists a real eigenvec-tor v ∈ Rn such that Bv = λ0v, as required. �

2.1.1. A geometric proof. The material in this subsection is optional.The second proof is somewhat longer but has the advantage of being moregeometric, as well as more concrete in the construction of the eigenvector.Let S ⊆ Rn be the unit sphere S = {v ∈ Rn : |v| = 1}. Given a sym-metric matrix B, define a function f(v) = 〈v,Bv〉. We are interested inits restriction to S, i.e., f : S → R. Let v0 be a maximum of f restrictedto S. We will show that v0 is an eigenvector of B. Let V ⊥

0 ⊆ Rn be theorthogonal complement of the line spanned by v0. Let w ∈ V ⊥

0 . Considerthe curve v0+ tw, t ≥ 0 (see also a different choice of curve in Remark 2.1.3below). Then d

dt

∣∣t=0

f(v0 + tw) = 0 since v0 is a maximum and w is tangentto the sphere. Now

d

dt

∣∣∣0f(v0 + tw) =

d

dt

∣∣0〈v0 + tw,B(v0 + tw)〉

=d

dt

∣∣0(〈v0, Bv〉+ t〈v0, Bw〉+ t〈w,Bv0〉+ t2〈w,Bv〉)

= 〈v0, Bw〉+ 〈w,Bv0〉= 〈Bw, v0〉+ 〈Bv0, w〉= 〈Btv0, w〉+ 〈Bv0, w〉= 〈(Bt +B)v0, w〉= 2〈Bv0, w〉 by symmetry of B.

2.2. TRACE OF PRODUCT OF MATRICES IN INDEX NOTATION 23

Thus 〈Bv0, w〉 = 0 for all w ∈ V ⊥0 . Hence Bv0 is proportional to v0 and

so v0 is an eigenvector of B.

Remark 2.1.3. Our calculation used the curve v0 + tw which, whiletangent to S at v0 (see section 6.8), does not lie on S. Therefore one needsto use instead the curve (cos t)v0 + (sin t)w lying on S. Then

d

dt

∣∣∣t=0

〈(cos t)v0 + (sin t)w,B((cos t)v0 + (sin t)w)〉 = · · · = 〈(Bt +B)v0, w〉and one argues by symmetry as before. Alternatively, one could note thatthe derivative is independent of the choice of curve representing the vectorand therefore the original choice of linear curve is valid, as well.

2.2. Trace of product of matrices in index notation

First we reinforce the material on index notation and Einstein sum-mation convention defined in Section 1.3. The following result is impor-tant in its own right. We reproduce it here because its proof is a goodillustration of the uses of the Einstein index notation. Recall (Defini-tion 1.5.2) that the trace of a square matrix A = (aij) is tr(A) = akk.Here k is a summation index (any other letter could have been used inplace of k).

Theorem 2.2.1. Square matrices A and B of the same size sat-isfy tr(AB) = tr(BA).

Proof. Let A = (aij) and B = (bij). Then

AB = (aikbkj) i=1,...,nj=1,...,n

By definition of trace (see Definition 1.5.2), we obtain

tr(AB) = aikbki. (2.2.1)

Similarly,tr(BA) = tr(bkia

ij) = bkia

ik = aikb

ki. (2.2.2)

Comparing the outcomes of calculations (2.2.1) and (2.2.2) we concludethat tr(AB) = tr(BA). �

Exercise 2.2.2. A matrix A is called idempotent if A2 = A. Writethe idempotency condition in indices with Einstein summation conven-tion (without Σ’s), keeping track of free indices and internal summationindices.

Exercise 2.2.3. Matrices A and B are similar if there exists aninvertible matrix P such that AP − PB = 0. Write the similaritycondition in indices, as the vanishing of each (i, j)th coefficient of thedifference AP − PB.


2.3. Inner product spaces and self-adjoint endomorphisms

In Section 2.1 we worked with real matrices and showed that thesymmetry of a matrix guarantees the existence of a real eigenvector.In a more general situation where a natural basis is not available, asimilar statement holds for a special type of endomorphism of a realvector space with an inner product.

Definition 2.3.1. Let (V, 〈 , 〉) be a real inner product space. Anendomorphism B : V → V is called selfadjoint if one has

〈Bv,w〉 = 〈v,Bw〉 ∀v, w ∈ V. (2.3.1)

The following is a consequence of Theorem 2.1.2.

Corollary 2.3.2. Every selfadjoint endomorphism of a real innerproduct space admits a real eigenvector.

Proof. The selfadjointness was the relevant property in the proofof Theorem 2.1.2 on symmetric matrices. �

2.4. Orthogonal diagonalisation of symmetric matrices

Let (V, 〈 , 〉) be a real inner product space. Consider an endomor-phism E : V → V .

Definition 2.4.1. A subspace U ⊆ V is invariant under E ifE(U) ⊆ U .

In other words, for every x ∈ U , one has E(x) ∈ U , as well.

Definition 2.4.2. The orthogonal complement O ⊆ V of a sub-space U ⊆ V is defined by O = {x ∈ V : 〈x, u〉 = 0 ∀u ∈ U}.

Such a situation is represented by the formula

V = U +O.

Example 2.4.3. Let U ⊆ R3 be the (x, y)-plane. Then its orthog-onal complement is the z-axis: O = Re3, and R3 = U +O.

The following theorem is known from linear algebra.

Theorem 2.4.4. If U and O are orthogonal complements of eachother in V then dimV = dimU + dimO.

Example 2.4.5 (Generalisation of Example 2.4.3). The orthogonalcomplement of the plane

U = {(x, y, z) : ax+ by + cz = 0}in R3 is the line O spanned by the vector (a, b, c)t 6= 0.

2.4. ORTHOGONAL DIAGONALISATION OF SYMMETRIC MATRICES 25

Lemma 2.4.6. Let E : V → V be a selfadjoint endomorphism of aninner product space V . Suppose U ⊆ V is an E-invariant subspace.Then the orthogonal complement of U in V is also E-invariant.

Proof. Let w be orthogonal to U ⊆ V . Let u ∈ U be an arbitraryvector. Then by selfadjointness,

〈E(w), u〉 = 〈w,E(u)〉 = 0

since E(u) ∈ U by E-invariance. Therefore the vector E(w) is alsoorthogonal to the vector u. Since this is valid for all u ∈ U , thevector E(w) also belongs to the orthogonal complement of U . Hencethe orthogonal complement of U is E-invariant, as required. �

Definition 2.4.7. A real square matrix P is orthogonal if PP t = I,i.e., P−1 = P t.

Theorem 2.4.8. Every real symmetric matrix can be orthogonallydiagonalized.

Proof. The proof will use self-adjoint endomorphisms and invari-ant subspaces. A symmetric n × n matrix S ∈ Mn,n(R) defines aselfadjoint endomorphism SR of the real inner product space V = Rn

given by

SR : V → V, v 7→ Sv.

We will give a proof in four steps.Step 1. By Corollary 2.3.2, every selfadjoint endomorphism has a

real eigenvector v1 ∈ V , which we can assume to be a unit vector:

|v1| = 1.

Let λ1 ∈ R be its eigenvalue.Step 2. We inductively construct a sequence of nested invariant

subspaces as follows. Let V1 = V and set

V2 ⊆ V1

be the orthogonal complement of the line Rv1 ⊆ V1. Thus we have anorthogonal decomposition

V1 = Rv1 + V2.

By Lemma 2.4.6, the subspace V2 is invariant under the endomor-phism SR. Consider the restriction

SR⇂V2

of SR to V2. The restriction SR⇂V2 is still selfadjoint by inheriting theproperty (2.3.1) restricted to V2. Namely, since property (2.3.1) holds


for all vectors v, w ∈ V , it still holds if these vectors are constrained tovary in a subspace V2 ⊆ V . Thus we have

〈SRv, w〉 = 〈v, SRw〉 ∀v, w ∈ V2. (2.4.1)

Therefore we can apply Corollary 2.3.2 to the selfadjoint endomor-phism SR⇂V2 of V2. As before, we find an eigenvector v2 ∈ V2 of SR⇂V2with eigenvalue λ2 ∈ R. Let V3 ⊆ V2 be the orthogonal complement ofthe line Rv2 spanned by v2. Next we choose an eigenvector v3 ∈ V3,etc. Arguing inductively, we obtain a strictly decreasing, or nested,2

sequence of spaces

V1 ⊃ V2 ⊃ V3 ⊃ . . . ⊃ Vi ⊃ . . . ⊃ {0}which necessarily ends with a point since V1 is finite-dimensional.

Step 3. We thus obtain an orthonormal basis consisting of uniteigenvectors v1, . . . , vn ∈ Rn belonging respectively to the eigenval-ues λ1, . . . , λn ∈ R of the endomorphism SR. Let

P = [v1 . . . vn]

be the orthogonal n × n matrix whose columns are the vectors vi, sothat we have

P−1 = P t. (2.4.2)

Step 4. Consider the diagonal matrix Λ = diag(λ1, . . . , λn). Byconstruction, we have

S = PΛP t

from (2.4.2), or equivalently,

SP = PΛ. (2.4.3)

Indeed, to verify the relation (2.4.3), note that both sides of (2.4.3) areequal to the square matrix [λ1v1 λ2v2 . . . λnvn]. �

2.5. Classification of conic sections: diagonalisation

We will now apply the linear-algebraic tools developed in the pre-vious sections in order to classify conic sections in the plane up toorthogonal transformations (rotations) and translations of the plane.

Definition 2.5.1. A conic section3 (or conic for short) in the planeis by definition a curve defined by the following master equation (gen-eral equation) in the (x, y)-plane:

ax2 + 2bxy + cy2 + dx+ ey + f = 0, a, b, c, d, e, f ∈ R. (2.5.1)

2mekunenet3chatach charut

2.5. CLASSIFICATION OF CONIC SECTIONS: DIAGONALISATION 27

Here we chose the coefficient of the xy term to be 2b rather than bso as to simplify formulas like (2.5.2) below.

Example 2.5.2. We have the following examples of conic sections:

(1) a circle x2 + y2 − 1 = 0,(2) an ellipse x2 + 3y2 − 1 = 0,(3) a parabola x2 + y = 0,(4) a hyperbola 2xy + 1 = 0,(5) a hyperbola x2 − 5y2 − 1 = 0.

Consider again the master equation (2.5.1). We now let X =

(xy

)

and let

S =

(a bb c

). (2.5.2)

Then X tSX = ax2 + 2bxy + cy2. We can now eliminate the mixedterm xy as follows.

Theorem 2.5.3. Up to an orthogonal transformation resulting innew coordinates (x′, y′), every conic section as in (2.5.1) can be writtenin a “diagonal” form

λ1x′ 2 + λ2y

′ 2 + d′x′ + e′y′ + f = 0, (2.5.3)

where the coefficients λ1 and λ2 are the eigenvalues of the matrix Sof (2.5.2).

Proof. We give a proof in four steps.Step 1. Consider the row vector

T =(d e

).

Then TX = dx+ ey. Thus equation (2.5.1) becomes

X tSX + TX + f = 0. (2.5.4)

Step 2. We apply Theorem 2.4.8 to orthogonally diagonalize thesymmetric matrix S to obtain S = PΛP t with

Λ =

(λ1 00 λ2

). (2.5.5)

Substituting this expression for S into (2.5.4) yields

X tPΛP tX + TX + f = 0.

Step 3. We set X ′ = P tX. Then X = PX ′ since P is orthogonal.Furthermore, we have

(X ′)t = (P tX)t = (X t)(P t)t = (X t)P.


Hence we obtain the equation

(X ′)tΛX ′ + TPX ′ + f = 0.

Step 4. Letting T ′ = TP , we obtain

(X ′)tΛX ′ + T ′X ′ + f = 0,

where Λ is the diagonal matrix of (2.5.5). Letting x′ and y′ be the

components of X ′, i.e. X ′ =

(x′

y′

), and T =

(d′ e′

), we obtain for-

mula (2.5.3), as required. �

Example 2.5.4. The hyperbola 2xy − 1 = 0 is not in “diagonal”

form. The corresponding matrix S is S =

(0 11 0

). Here the eigen-

values are λ1 = 1 and λ2 = −1. By Theorem 2.5.3 we obtain the“diagonal” equation

x′2 − y′2 − 1 = 0

in the new coordinates (x′, y′). The result could be obtained directly

by using the substitution x = x′+y′√2

and y = x′−y′√2.

To obtain more precise information about the conic, we need tospecify certain nondegeneracy conditions, as discussed in Section 2.6.

2.6. Classification of conics: trichotomy, nondegeneracy

We apply the diagonalisation result of Section 2.5 to classify conicsections into three types (under suitable nondegeneracy conditions):ellipse, parabola, hyperbola. Such a result can be referred to as tri-chotomy.4 Let S be the matrix (2.5.2).

Theorem 2.6.1. Suppose det(S) 6= 0, i.e., S is invertible. Then,up to an orthogonal transformation and a translation, the conic sectioncan be written in the form

λ1(x′′)2 + λ2(y

′′)2 + f ′′ = 0 (2.6.1)

Note that the coefficients λ1 and λ2 are the same as in (2.5.3) butthe constant term is changed from f to f ′′.

Proof. If the determinant is nonzero then both eigenvalues λ1, λ2are nonzero. The term d′x′ in (2.5.3) can be absorbed into the quadratic

4trichotomia

2.6. CLASSIFICATION OF CONICS: TRICHOTOMY, NONDEGENERACY 29

term λ1x′2 by completing the square as follows. We write

λ1x′2 + d′x′ = λ1

(x′2 + 2

d′

2λ1x′)

= λ1

(x′2 + 2

d′

2λ1x′ +

(d′

2λ1

)2)

− λ1

(d′

2λ1

)2

= λ1

(x′ +

d′

2λ1

)2

− d′2

4λ1.

Then we set

x′′ = x′ +d′

2λ1.

Similarly e′y′ can be absorbed into λ2y′2. Geometrically this corre-

sponds to a translation along the axes x′ and y′, proving the theo-rem. �

Definition 2.6.2. A conic section of type (2.6.1) is called a hy-perbola if λ1λ2 < 0, provided the following nondegeneracy condition issatisfied: the constant f ′′ in equation (2.6.1) is nonzero.

Corollary 2.6.3 (Degenerate case). Assume det(S) < 0. If theconstant f ′′ is zero, then instead of a hyperbola, the solution set is adegenerate conic given by a pair of transverse lines.

Remark 2.6.4. The transverse lines as in Corollary 2.6.3 are notnecessarily orthogonal. More specifically, they are orthogonal if andonly if λ1 = −λ2.

Definition 2.6.5. A conic section is called an ellipse if λ1λ2 > 0,provided the following nondegeneracy condition is satisfied: the con-stant f ′′ in equation (2.6.1) is nonzero and has the opposite sign ascompared to the sign of λ1, i.e., f

′′λ1 < 0.

See Example 11.5.6 for an application.

Corollary 2.6.6 (Degenerate case). Assume det(S) > 0.

(1) If the constant f ′′ is zero, then instead of an ellipse one obtainsa single point x′′ = y′′ = 0.

(2) If the constant f ′′ 6= 0 has the same sign as λ1 then the solutionset is empty.

We summarize our conclusions for the ellipse and the hyperbola inthe following corollary. Recall that the determinant of the matrix S isthe product of its eigenvalues: det(S) = ac− b2 = λ1λ2.


Corollary 2.6.7 (Case det(S) > 0). We have the following rela-tions between the algebraic equation and the nature of the correspondingconic:

(1) If the conic ax2 + 2bxy + cy2 + dx + ey + f = 0 is an ellipsethen ac− b2 > 0.

(2) If ac − b2 > 0 and the solution locus is neither empty nor asingle point, then it is an ellipse.

Corollary 2.6.8 (Case det(S) < 0). We have the following rela-tions between the algebraic equation and the nature of the correspondingconic:

(1) If the conic ax2 + 2bxy + cy2 + dx+ ey + f = 0 is a hyperbolathen ac− b2 < 0.

(2) If ac− b2 < 0 and the solution locus is not a pair of transverselines, then the conic is a hyperbola.

The remaining case of vanishing determinant will be treated in Sec-tion 2.7.

2.7. Characterisation of parabolas

Suppose the matrix S of (2.5.2) satisfies det(S) = 0. In this case,one cannot necessarily eliminate the linear terms by completing thesquare as in the cases of ellipse and hyperbola treated in Section 2.6.Therefore we continue working with the equation

λ1x′2 + λ2y

′2 + d′x′ + e′y′ + f = 0 (2.7.1)

from (2.5.3) where now one of the λi is zero. This is legitimate becausediagonalizing S does not depend on S being invertible and only requiressymmetry.

Definition 2.7.1. The conic (2.7.1) is a parabola if the followingtwo conditions are satisfied by the coefficients in (2.7.1):

(1) the matrix S is of rank 1 (equivalently, λ1λ2 = 0 and λ1 6= λ2);(2) either λ1e

′ 6= 0 or λ2d′ 6= 0.

The following corollary describes the remaining degenerate cases.

Corollary 2.7.2 (Degenerate case). Suppose S is of rank 1 andafter absorbing the linear term in (2.7.1) the equation becomes λ1x

′′2+f ′′ = 0 or λ2y

′′2+f ′′ = 0. Then the solution set is either empty, a line,or a pair of parallel lines.

Example 2.7.3. The equation x2 + 1 = 0 has an empty solutionset in the (x, y)-plane. The equation x2 = 0 has as solution set a single

2.7. CHARACTERISATION OF PARABOLAS 31

line, namely the y-axis. The solution set of the equation x2 − 1 = 0 isa pair of parallel lines.

Remark 2.7.4. Every nondegenerate conic (ellipse, hyperbola, pa-rabola) given by the solution set of the equation

ax2 + 2bxy + cy2 + dx+ ey + f = 0

can in fact be represented (up to orthogonal transformation) by theintersection of a suitable plane in R3 and the standard cone with equa-tion z2 = x2 + y2. However, some degenerate cases cannot be repre-sented by such an intersection. For example, the pair of parallel linesoccurring in Corollary 2.7.2 cannot be represented as the intersectionof a plane and the cone (neither can the empty set).

CHAPTER 3

Quadric surfaces, Hessian, representation of curves

Before we go on to quadric (i.e., quadratic) surfaces, we summarizethe results obtained for quadratic curves in the previous chapter.

3.1. Summary: classification of quadratic curves

The analysis of the cases presented in Sections 2.6 and 2.7 resultsin the following classification.

Theorem 3.1.1. Let a, b, c, d, e, f ∈ R, and assume not all of a, b, care 0. A real quadratic curve section ax2+2bxy+cy2+dx+ey+f = 0is represented by one of the following possible sets:

(1) the empty set ∅;(2) a single point;(3) union of a pair of transverse lines;(4) a single line or a pair of parallel lines;(5) ellipse (and then ac− b2 > 0);(6) parabola (and then ac− b2 = 0);(7) hyperbola (and then ac− b2 < 0).

Remark 3.1.2. The first four cases are known as degenerate cases.A parabola can occur only if ac − b2 = 0. An ellipse can occur onlyif ac− b2 > 0. A hyperbola can occur only if ac− b2 < 0.

3.2. Quadric surfaces

Quadric surfaces1 are a rich source of examples that will help usillustrate basic notions of differential geometry such as Gaussian cur-vature.

Definition 3.2.1. A quadric surfaceM ⊆ R3 is the locus of points(x, y, z) satisfying the master equation

ax2+2bxy+ cy2+2dxz+fz2+2gyz+hx+ iy+ jz+k = 0, (3.2.1)

where a, b, c, d, f, g, h, i, j, k ∈ R.

1mishtachim ribu’im

33

34 3. QUADRIC SURFACES, HESSIAN, REPRESENTATION OF CURVES

To bring equation (3.2.1) to standard form, we apply an orthogo-nal diagonalisation procedure similar to that employed in Section 2.6.Thus, we define matrices S, X, and D by setting

S =

a b db c gd g f

, X =

xyz

, D =

(h i j

), (3.2.2)

so that the quadratic part of (3.2.1) becomes X tSX, and the linearpart becomes DX. Then equation (3.2.1) takes the form

X tSX +DX + k = 0. (3.2.3)

Theorem 3.2.2. By means of an orthogonal transformation, thegeneral equation (3.2.1) of a quadric surface M can be simplified to

λ1x2 + λ2y

2 + λ3z2 + dx+ fy + gz + k = 0, (3.2.4)

with new variables x, y, z and new coefficients λ1, λ2, λ3, d, f, g, h ∈ R,and the same k as in (3.2.3), where λi, i = 1, 2, 3 are the eigenvaluesof S.

Proof. We orthogonally diagonalize the symmetric matrix S as inSection 2.5. �

3.3. Case of eigenvalues (+ + +) or (−−−), ellipsoid

Definition 3.3.1. A quadric surface M is an ellipsoid if

(1) the coefficients λi, i = 1, 2, 3 in (3.2.4) are all nonzero (i.e., det(S) 6=0);

(2) all three have the same sign; and(3) the solution locus is neither a single point nor the empty set.

Theorem 3.3.2. Suppose the eigenvalues of S in the master equa-tion (3.2.3) of a quadric surfaceM are nonzero and have the same sign.Then an orthogonal transformation and translation of the coordinatesreduce the equation to the form

λ1x2 + λ2y

2 + λ3z2 + ℓ = 0,

for new variables x, y, z. The following three cases can then occur forthe conic:

(1) an ellipsoid if λ1ℓ < 0;(2) a degenerate surface given by a single point if ℓ = 0;(3) the empty set when λ1ℓ > 0.

3.4. DETERMINATION OF TYPE OF QUADRIC SURFACE: EXPLICIT EXAMPLE35

Proof. By Theorem 3.2.2 there exists an orthogonal transforma-tion diagonalizing S. Next, we use the nonvanishing of the eigenvaluesto complete the square as in Section 2.6, so as to eliminate the first-order term D in equation (3.2.3). �

Example 3.3.3. We have the following examples of ellipsoids:

(1) the equation 3x2 + 5y2 + 7z2 − 1 = 0 defines an ellipsoid;(2) equivalently −3x2 − 5y2 − 7z2 +1 = 0 gives the same ellipsoid

as in (1);(3) the equation 3x2 +5y2 +7z2 = 0 degenerates to a single point

(the origin),(4) equation 3x2 + 5y2 + 7z2 + 1 = 0 is the empty degenerate

quadric surface.

3.4. Determination of type of quadric surface: explicitexample

To present an application of Theorem 3.3.2, let us calculate out anexplicit example. Consider the surfaceM ⊆ R3 defined by the equation

3x2 + y2 − 2xz + 3z2 − 5 = 0. (3.4.1)

Let us determine the type of surface it is.Step 1. The corresponding symmetric matrix S ∈ Mat3,3(R) is

S =

3 0 −10 1 0−1 0 3

.

Step 2. Note that the equation is partly diagonalized already be-cause there are no xy or yz terms. Thus the y-axis is an invariantsubspace of S. Namely, the y-axis is the eigenspace of the eigen-value λ1 = +1. Its orthogonal complement, the (x, z)-plane, is alsoinvariant under S by Lemma 2.4.6, or simply by inspection.

Step 3. To find the remaining two eigenvalues of S, we restrict theendomorphism to the (x, z) plane. Here we obtain the equation

3x2 − 2xz + 3z2 − 5 = 0. (3.4.2)

We therefore focus on diagonalizing the quadratic part 3x2− 2xz+3z2

of (3.4.2). The corresponding symmetric matrix is

C =

(3 −1−1 3

)

with characteristic polynomial pC(λ) = λ2 − 6λ + 8. Its roots are 3 ±√9− 8 = 3± 1. Both roots λ2 = 2 and λ3 = 4 are positive.


Step 4. According to the general theory in dimension 2, we obtainthat, after an orthogonal transformation and with respect to the newcoordinates x′, z′, equation (3.4.2) becomes

2x′2 + 4z′2 − 5 = 0.

Since we are only interested in the eigenvalues, there is no need todetermine x′, z′ explicitly (which would involve calculating the eigen-vectors).

Step 5. In the new coordinates (y, x′, z′), the equation of thequadric surface M of (3.4.1) takes the form

y2 + 2x′ 2 + 4z′ 2 − 5 = 0.

Notice that all three eigenvalues 1, 2, 4 of the original matrix S arepositive. Furthermore, the solution set is neither a point nor the emptyset since the constant term ℓ = −5 is negative. By Theorem 3.3.2 (orDefinition 3.3.1), M is an ellipsoid.

3.5. Case of eigenvalues (+ +−) or (+−−), hyperboloid

Let S be a symmetric matrix as in (3.2.2) with eigenvalues λi, i =1, 2, 3. As in Section 3.4 we assume that detS = λ1λ2λ3 6= 0.

Definition 3.5.1. Suppose λ1, λ2, λ3 include both positive andnegative values. Then

(1) The solution set of the homogeneous equation

λ1x2 + λ2y

2 + λ3z2 = 0 (3.5.1)

is called a cone;(2) A surface defined by an equation reducible to (3.5.1) by an

orthogonal transformation and translation is similarly called acone.

Example 3.5.2. The equation x2 + y2 − z2 = 0 defines a cone, andsimilarly the equation −x2 − y2 + z2 = 0 defines (the same) cone.

Definition 3.5.3. A quadric surface M is a hyperboloid if the co-efficients λ1, λ2, λ3 in (3.2.4) include both positive and negative values,and M is not a cone.

Theorem 3.5.4. Suppose det(S) 6= 0 in the master equation of aquadric surface M .

(1) If all eigenvalues have the same sign and M is neither theempty set nor single point, then M is an ellipsoid;

(2) if the eigenvalues include both positive and negative valuesand M is not a cone, then M is a hyperboloid.

3.6. CASE RANK(S) = 2; PARABOLOID, HYPERBOLIC PARABOLOID 37

Proof. We apply orthogonal diagonalisation as before. �

One distinguishes two types of hyperboloids as follows.

Definition 3.5.5. The hyperboloid of one sheet2 is the locus of theequation

az2 = bx2 + cy2 − d

where a > 0, b > 0, c > 0, d > 0 (or all negative).

Remark 3.5.6. The Gaussian curvature (see Section 9.7) of a hy-perboloid of one sheet is negative at each point.

Definition 3.5.7. The hyperboloid of two sheets3 is the locus ofthe equation

az2 = bx2 + cy2 + d

where a > 0, b > 0, c > 0, d > 0 (or all negative).

Remark 3.5.8. The Gaussian curvature of a hyperboloid of twosheets is positive at each point.

Remark 3.5.9. In this dimension (unlike the case of conics), thesign of the determinant det(S) has no significance and cannot be usedto determine whether the surface M is an ellisoid or a hyperboloid, orwhich type of hyperboloid M is.

3.6. Case rank(S) = 2; paraboloid, hyperbolic paraboloid

We have rank(S) = 2 if exactly two of the three eigenvalues arenonzero. We study quadric surfaces

X tSX +DX + k = 0 (3.6.1)

in the case when S has rank 2.

Theorem 3.6.1. Suppose matrix S in (3.6.1) has rank(S) = 2. Upto orthogonal transformation and translation, the quadric surface Mdefined by X tSX +DX + k = 0 takes the form

λ1x2 + λ2y

2 + cz + d = 0, (3.6.2)

with new variables x, y, z and new coefficients c, d, where the eigenval-ues satisfy λ1λ2 6= 0.

Proof. We orthogonally diagonalize the matrix as before. Werelabel the new coordinates as x, y, z in such a way that

(1) the eigendirections for the nonzero eigenvalues correspond tothe x-axis and the y-axis, and

2chad-yeriati3du-yeriati


(2) the zero eigenvalue corresponds to the z-axis.

Next, we eliminate the linear terms in x and y by absorbing it intothe respective quadratic term by completing the square as before. �

Remark 3.6.2. Note that the linear term in z cannot be eliminatedbecause there is no quadratic term to absorb it into, since the thirdeigenvalue vanishes.

Definition 3.6.3. If c 6= 0 in equation (3.6.2), the correspondingnondegenerate quadric surface M is called a paraboloid.

Additional special cases of quadric surfaces are the following. It isimportant to think through each of these examples as they will provideimportant illustrations of the behavior of the Gaussian curvature (tobe introduced in Section 9.7) of surfaces.

Definition 3.6.4. The (elliptic) paraboloid z = ax2 + by2 + d,where ab > 0.

Here the Gaussian curvature is positive at each point.

Definition 3.6.5. The hyperbolic paraboloid is the surface z =ax2 − by2, where ab > 0.

Here the Gaussian curvature is negative at each point. See Fig-ures 3.6.1, 3.6.2, 3.6.3.

Figure 3.6.1. Hyperbolic paraboloid

Corollary 3.6.6. Consider the surface

M = {(x, y, z) ∈ R3 : z = ax2 + by2} where ab 6= 0.

If ab > 0 then M is a (elliptic) paraboloid. If ab < 0 then M is ahyperbolic paraboloid.

This is immediate from the discussion above.

3.7. CYLINDER 39

Figure 3.6.2. Elliptic and hyperbolic paraboloids

Figure 3.6.3. Hyperbolic paraboloid

3.7. Cylinder

Definition 3.7.1. Suppose c = 0 in equation (3.6.2), resultingin equation λ1x

2 + λ2y2 + d = 0. All such surfaces in R3 are called

degenerate.

Example 3.7.2. The cylinder x2 + y2 − 1 = 0 is an example of adegenerate quadric surface.


Remark 3.7.3. In Section 3.1 we gave a (more-or-less) completeclassification of conic sections, including the degenerate cases. Forquadric surfaces, a complete classification in the case det(S) = 0 is toodetailed to be treated here.

3.8. Exercise on index notation

Theorem 3.8.1. Every 2× 2 matrix A = (aij) satisfies the identity

(∀i, j) aikakj + det(A) δij = akka

ij (3.8.1)

Note that indices i and j and free indices, whereas k is a summationindex (see Section 1.8).

Proof. Let pA(λ) be the characteristic polynomial of A. For 2× 2matrices, we have pA(λ) = λ2− (trA)λ+det(A). The Cayley-Hamiltontheorem asserts that pA(A) = 0. Therefore we obtain

A2 − (trA)A+ det(A)I = 0,

which in index notation gives

aikakj − akka

ij + detA δij = 0.

This is equivalent to (3.8.1). �

Example 3.8.2 (Exercise on index notation). For a matrix A ofsize 3× 3, the characteristic polynomial pA(λ) has the form

−pA(λ) = λ3 − Tr(A)λ2 + s(A)λ− det(A)λ0.

Here s(A) = λ1λ2+λ1λ3+λ2λ3, where λi are the eigenvalues of A. ByCayley-Hamilton theorem, we have

pA(A) = 0. (3.8.2)

Express the equation (3.8.2) in index notation.

3.9. Gradient and Hessian

We review some material from vector calculus. Let (u1, . . . , un) becoordinates in Rn. Consider a real-valued function f(u1, . . . , un) of nvariables. The function will be required to be at least C2-smooth.

Definition 3.9.1. The gradient of f at a point p = (u1, . . . , un) isthe vector

∇f(p) =

∂f∂u1∂f∂u2...∂f∂un

3.10. MINIMA, MAXIMA, SADDLE POINTS 41

Definition 3.9.2. A critical point p ∈ Rn of f is a point satisfying

∇f(p) = 0,

i.e. ∂f∂ui

(p) = 0 for all i = 1, . . . , n.

We now consider the second partial derivatives of f , which we will

denote fij =∂2f

∂ui∂uj.

Definition 3.9.3. The Hessian matrix of f

Hf =(fij)i=1,...,nj=1,...,n

is the n× n matrix of second partial derivatives.

The antisymmetrisation notation was defined in formula (1.6.2).We can then formulate what is known as Schwarz’s theorem or Clairaut’stheorem as follows.

Theorem 3.9.4 (Equality of mixed partials). Let f ∈ C2. Thefollowing three statements are equivalent and true:

(1) the Hessian Hf is a symmetric matrix;(2) we have fij = fji for all i, j;(3) in terms of the antisymmetrisation notation, f[ij] = 0.

3.10. Minima, maxima, saddle points

We study the graph of a function f(u1, u2) at a critical point.

Example 3.10.1 (maxima, minima, saddle points). Let n = 2.Then the sign of the determinant

det(Hf ) =∂2f

∂u1∂u1∂2f

∂u2∂u2−(

∂2f

∂u1∂u2

)2

at a critical point p has geometric significance. Namely, if

det(Hf (p)) > 0,

then p is a local maximum or a local minimum. If det(Hf (p)) < 0,then p is a saddle point4 of the graph of the function.

Example 3.10.2 (Quadric surfaces). Quadric surfaces are a richsource of examples.

(1) The origin is a critical point for the function whose graph isthe paraboloid z = x2 + y2. In the case of the paraboloid thecritical point is a minimum.

4Nekudat ukaf


(2) Similar remarks apply to the top sheet of the hyperboloid of

two sheets, namely z =√x2 + y2 + 1, where we also get a

minimum.(3) The origin is a critical point for the function whose graph is

the hyperbolic paraboloid z = x2 − y2. In the case of thehyperbolic paraboloid the critical point is a saddle point.

In addition to the sign, the value of Hf (p) also has geometric signif-icance in terms of an invariant we will define later called the Gaussiancurvature, expressed by the following theorem that will be proved later.

Theorem 3.10.3. Let f be a function of two variables. Considerthe surface M ⊆ R3 given by the graph of f in R3. Let p ∈ R2 bea critical point of f . Then the value of det(Hf (p)) is precisely theGaussian curvature of the surface M at the point (p, f(p)) ∈ R3.

See Definition 9.7.1 for more details. We will return to surfaces inSection 5.2.

3.11. Parametric representation of a curve

In Chapter 2 we studied curves given by solution sets of quadraticequations in the plane. We will now study more general curves.

Remark 3.11.1. There are two main ways of representing a curvein the plane: parametric and implicit.

A curve in the plane can be represented by a pair of coordinatesevolving as a function of the time parameter t:

α(t) = (α1(t), α2(t)), t ∈ [a, b],

with coordinates α1(t) and α2(t). Both functions are assumed to be ofclass C2. Thus a parametrized curve can be viewed as a map

α : [a, b] → R2. (3.11.1)

Let C be the image of the map (3.11.1). Then C ⊆ R2 is the geometriccurve independent of parametrisation. Thus, changing the parametri-sation by setting t = t(s) and replacing α by a new curve β(s) = α(t(s))preserves the geometric curve C.

Definition 3.11.2. A parametrisation α(t) is called regular if forall t one has α′(t) 6= 0.

3.13. IMPLICIT FUNCTION THEOREM 43

3.12. Implicit representation of a curve

A curve in the (x, y)-plane can also be represented implicitly as thesolution set of an equation

F (x, y) = 0,

where F is a function always assumed to be of class C2(R2).

Definition 3.12.1. The level curve CF ⊆ R2 is the locus of theequation

CF = {(x, y) : F (x, y) = 0}.Example 3.12.2. A circle of radius r > 0 corresponds to the choice

of the function F (x, y) = x2 + y2 − r2.

Further examples are given below.

(1) The function F (x, y) = y − x2 defines a parabola.(2) The function F (x, y) = xy − 1 defines a hyperbola.(3) The function F (x, y) = x2 − y2 − 1 also defines a hyperbola.

Remark 3.12.3. In each of these cases, it is easy to find a parametri-sation (at least of a part of) the level curve, by solving the equationfor one of the variables. Thus, in the case of the circle, we choose thepositive square root to obtain y =

√r2 − x2, giving a parametrisation

of the upper half-circle by means of the pair of formulas

α1(t) = t, α2(t) =√r2 − t2.

Note this is not all of the level curve CF .

Remark 3.12.4. Unlike the above examples, typically it is difficultto find an explicit parametrisation for a curve defined by an implicitequation F (x, y) = 0. Locally one can always find one in theory undera suitable nondegeneracy condition, expressed by the implicit functiontheorem, dealt with in Section 3.13.

3.13. Implicit function theorem

Theorem 3.13.1 (implicit function theorem). Assume the func-tion F (x, y) is C2(R2). Suppose the gradient of F does not vanish at aspecific point p = (x, y) ∈ CF , in other words

∇F (p) 6= 0.

Then there exists a regular parametrisation (α1(t), α2(t)) of the levelcurve CF in a suitable neighborhood of p.

A useful special case is the following result.


Theorem 3.13.2 (implicit function theorem: special case). Assumethe function F (x, y) is C2. Suppose that the partial derivative withrespect to y satisfies

∂F

∂y(p) 6= 0.

Then there exists a parametrisation y = α2(x), in other words α(t) =(t, α2(t)) of the curve CF in a suitable neighborhood of p.

Example 3.13.3. In the case of the circle x2+y2 = r2, the point (r, 0)on the x-axis fails to satisfy the hypothesis of Theorem 3.13.2. Thecurve cannot be represented by a differentiable function y = y(x) in aneighborhood of this point due to the problem of a vertical tangent. Onthe other hand, a regular parametrisation still exists, e.g., (r cos t, r sin t).

3.13.1. Jacobi’s criterion. This section is optional. The type ofquadric surface one obtains depends critically on the signs of the eigenvaluesof the matrix S. The signs of the eigenvalues can be determined withoutdiagonalisation by means of Jacobi’s criterion. Given a matrix A over afield F , let ∆k denote the k × k upper-left block, called a principal minor.Matrices A and B are equivalent if they are congruent (rather than similar),meaning that B = P tAP (rather than by conjugation P−1AP ). Equivalentmatrices don’t have the same eigenvalues (unlike similar matrices). Here Bis of course not assumed to be orthogonal.

Theorem 3.13.4 (Jacobi). Let A ∈Mn(F ) be a symmetric matrix, andassume det(∆k) 6= 0 for k = 1, . . . , n. Then A is equivalent to the matrix

diag(

1∆1, ∆1

det∆2, . . . , det∆n−1

det∆n

).

For example, for a 2 × 2 symmetric matrix

(a bb d

)Jacobi’s criterion

affirms the equivalence to

(1a 00 a

ad−b2

).

Proof. Take the vector bk = ∆−1k ek ∈ F k of length k, and pad it with

zeros up to length n. Consider the matrix B = (bij) whose column vectorsare b1, . . . , bn. By Cramer’s formula, the diagonal coefficients of B sat-

isfy bkk = det

(∆k−1 00 1/det∆k

)= det∆k−1/∆k, so det(B) =

∏nk=1 bkk =

1/det(A) 6= 0. Compute that BtAB is lower triangular with diagonalb11, . . . , bkk. Being symmetric, it is diagonal. �

If some minor ∆k is not invertible, then A cannot be definite. Ap-plying this result in the case of a real symmetric matrix, we obtain thefollowing corollary. Let A be symmetric. Then A is positive definite ifand only if all det(∆k) > 0. Define minors in general (choose rows and

3.13. IMPLICIT FUNCTION THEOREM 45

columns i1, . . . , it). Permuting rows and columns, we obtain the followingcorollary.

Corollary 3.13.5. Let A be symmetric positive definite matrix. Thenall “diagonal” minors are positive definite (and in particular have positivedeterminants).

An application of this is determining whether or not a quadric surfaceis an ellipsoid, without having to orthogonally diagonalize the matrix ofcoefficients, as we will see below (care has to be taken to show that thequadric surface is nondegenerate). For example, let us determine whetheror not the quadric surface

x2 + xy + y2 + xz + z2 + yz + x+ y + z − 2 = 0 (3.13.1)

is an ellipsoid. To solve the problem, we first construct the corresponding

matrix S =

1 1/2 1/21/2 1 1/21/2 1/2 1

and then calculate the principal minors ∆1 =

1, ∆2 = 1·1− 12 · 12 = 3/4, and ∆3 = 1+ 1

8+18− 1

4− 14− 1

4 = 12 . Thus all principal

minors are positive and therefore the surface is an ellipsoid, provided we can

show it is nondegenerate. To check nondegeneracy, notice that (3.13.1) has

at least two distinct solutions: (x, y, z) = (1, 0, 0) and (x, y, z) = (0, 1, 0).

Therefore it is a nondegenerate ellipsoid.

CHAPTER 4

Curvature, Laplacian, Bateman–Reiss

4.1. Unit speed parametrisation

In Section 3.11 we dealt with the notion of a parametrized curve.Let us review the notation involved.

• α : I → R2, α(t) = (α1(t), α2(t)) is a parametrized curve inthe plane;

• C ⊆ R2 is the underlying geometric curve, i.e., the image of α:

C ={(x, y) ∈ R2 : (∃t)

(x = α1(t), y = α2(t)

)}.

Definition 4.1.1. We say α = α(t) is a unit speed curve if∣∣dαdt

∣∣ = 1,

i.e.(dα1

dt

)2+(dα2

dt

)2= 1 for all t ∈ [a, b].

Definition 4.1.2. The parameter of a unit speed parametrisation,usually denoted s, is called arclength.

Similarly, a parametrized curve α(t) in Rn is unit speed if |α′(t)| = 1for all t.

Example 4.1.3 (Circle of radius r). Let r > 0. Then the curve

α(s) =(r cos

s

r, r sin

s

r

)

is a unit speed (arclength) parametrisation of the circle of radius r.Indeed, we have∣∣∣∣dα

ds

∣∣∣∣ =

√(r1

r

(− sin

s

r

))2

+

(r1

rcos

s

r

)2

=

√sin2 s

r+ cos2

s

r= 1.

A regular curve always admits an arclength parametrisation; seeTheorem 5.1.

4.2. Geodesic curvature of a curve

We review the standard calculus topic of the curvature of a curve.

Remark 4.2.1. The notion of the curvature of curves is closelyrelated to the theory of surfaces in Euclidean space. It is indispens-able to understanding the principal curvatures of a surface; see e.g.,Theorem 9.9.1 and Theorem 9.11.5.

47

48 4. CURVATURE, LAPLACIAN, BATEMAN–REISS

Our main interest will be in space curves (curves in R3). For suchcurves, one cannot in general assign a sign to the curvature. Thereforein the definition below we do not concern ourselves with the sign of thecurvature of plane curves, either (see Remark 4.2.3). Signed curvaturewill be defined in Section 10.8.4.

In the present section, only local properties of the curvature ofcurves will be studied.

Remark 4.2.2 (Signed curvature). For oriented plane curves, thereis a finer invariant called signed curvature; see Definition 10.8.4. Aglobal result on the curvature of plane curves appears in Section 11.5.

Definition 4.2.3. The (geodesic) curvature function kα(s) ≥ 0 ofa unit speed curve α(s) is the function

kα(s) =

∣∣∣∣d2α

ds2

∣∣∣∣. (4.2.1)

Theorem 4.2.4. The curvature kα(s) of the circle of radius r > 0is kα(s) =

1rat every point of the circle.

Proof. With the parametrisation given in Example 4.1.3, we have

d2α

ds2=

(r1

r2

(− cos

s

r

), r

1

r2

(− sin

s

r

))

for all s, and so kα =∣∣1r

(− cos s

r,− sin s

r

)∣∣ = 1r. �

Remark 4.2.5. For the circle, the curvature is independent of thepoint, i.e. is a constant function of s.

In Section 11.1, we will give a formula for curvature with respectto an arbitrary parametrisation (not necessarily arclength).

In Section 11.2, the curvature will be expressed in terms of theangle θ formed by the tangent vector (to the curve) with the positive x-axis.

4.3. Tangent and normal vectors

Consider a plane curve C ⊆ R2 parametrized by α(s) = (α1(s), α2(s))where s is the arclength.

Definition 4.3.1. The unit tangent vector v(s) at a point α(s)of C is the vector

v(s) = α′(s),

written in coordinates as v(s) = (x(s), y(s)).

Remark 4.3.2. x(s) and y(s) are the coordinates of v(s) rather

than of α(s). Thus x(s) = dα1

ds, y(s) = dα2

ds.

4.3. TANGENT AND NORMAL VECTORS 49

Definition 4.3.3. The unit normal vector n(s) to the curve C ⊆R2 is the vector

n(s) = (y(s),−x(s)).Remark 4.3.4 (Gauss map). Let S1 = {(x, y) : x2 + y2 = 1} be

the unit circle. Since |n(s)| = 1 by definition, one can think of n(s)as a map C → S1. This leads us to the notion of Gauss map for thecurve C; see Section 11.2.

Lemma 4.3.5. Let v(s) = (x(s), y(s)) be the unit tangent vector ofthe curve α(s), and assume y(s) 6= 0. Then

dy

ds= −dx

ds

x

y. (4.3.1)

Proof. We start with the formula v(s) = (x(s), y(s)). Differenti-ating the identity x(s)2 + y(s)2 = 1 with respect to s using chain rule,we obtain xdx

ds+ y dy

ds= 0 and therefore dy

ds= −dx

dsxyas required. �

Lemma 4.3.6. Let v(s) = (x(s), y(s)) be the unit tangent vectorof α(s), and assume y(s) 6= 0. With respect to the arclength parame-ter s, we have the following formula for the curvature:

kα(s) =

∣∣∣∣1

y

dx

ds

∣∣∣∣ .

Proof. We use Lemma 4.3.5. Substituting the expression (4.3.1)into α′′ = v′ we obtain

α′′(s) =

(dx

ds,dy

ds

)

=

(dx

ds,−dx

ds

x

y

)

=dx

ds

(1,−x

y

)

=1

y

dx

ds(y,−x)

=

(1

y

dx

ds

)n

and therefore kα(s) = |α′′(s)| =∣∣∣ 1y dxds

∣∣∣, as required. �

Proposition 4.3.7. Differentiating the normal vector

n(s) = (y(s),−x(s))


along the curve produces a vector proportional to v(s) with factor ofproportionality (up to sign) given by the geodesic curvature kα(s):

d

dsn(s) = ±kα(s)v(s). (4.3.2)

Proof. Differentiating and applying (4.3.1), we obtain

−n′ (s) =

(−dyds,dx

ds

)

=

(x

y

dx

ds,dx

ds

)

=dx

ds

(x

y, 1

)

=1

y

dx

ds(x, y)

=

(1

y

dx

ds

)v(s),

and we apply Lemma 4.3.6 to prove the proposition. �

Remark 4.3.8 (Comparison to the Weingarten map). A similarphenomenon to (4.3.2) (relating curvature to the derivative of the nor-mal) occurs for a surface M . In the case of surfaces, the directionalderivative of the normal vector n toM is used to define the Weingartenmap (shape operator). The latter leads to the Gaussian curvature func-tion K of M ; see Section 8.11.

4.4. Osculating circle of a curve

To give a more geometric description of the curvature of a curve, wewill exploit its osculating circle1 at a point. We first recall the followingfact from elementary calculus about the second derivative of a function.

Theorem 4.4.1. Let I be an interval, and let f ∈ C2(I). Thesecond derivative of a function f(x) may be computed from a triple ofpoints f(x), f(x+h), f(x−h) that are infinitely close to each other, asfollows:

f ′′(x) = limh→0

f(x+ h) + f(x− h)− 2f(x)

h2.

The theorem remains valid for vector-valued functions; see Defini-tion 4.4.3.

1Maagal noshek

4.5. CENTER OF CURVATURE, RADIUS OF CURVATURE 51

Corollary 4.4.2. The second derivative of a function can be cal-culated from the value of the function at a triple of nearby points x, x+h, x− h.

Definition 4.4.3. The osculating circle to the curve parametrizedby α with arclength parameter s at a point p = α(s) is obtained bychoosing a circle passing through the three points

α(s), α(s− h), α(s+ h)

for infinitesimal h, i.e., taking the limit as h tends to zero.

Remark 4.4.4. The osculating circle and the curve are “betterthan tangent” at the point p, in the sense that they have second ordertangency at p.

Corollary 4.4.5. The curvatures of the osculating circle and thecurve at the point of tangency are equal.2

Proof. The curvature is defined by the second derivative. Thesecond derivative is computed from the same triple of points for α(s)and for the osculating circle (see Corollary 4.4.2), proving the corollary(cf. Weatherburn [We55, p. 13]). �

4.5. Center of curvature, radius of curvature

Recall that we denote the geometric curve by C ⊆ R2 and itsparametrisation by α(s) where s is the arclength parameter.

To help geometric intuition, it is useful to recall Leibniz’s andCauchy’s definition of the radius of curvature of a curve [Cauchy 1826]as the distance from the curve to the intersection point of two infinitelyclose normals to the curve. In more detail, we have the following.

Definition 4.5.1. Let p ∈ C ⊆ R2. The center of curvature q of Cassociated with p is the intersection point of the normals to the curveat infinitely close points p and p′ of C.

2For example, let y = f(x). Compute the curvature of the graph of fwhen f(x) = ax2. Let B = (x, x2). Let A be the midpoint of OB. Let C be theintersection of the perpendicular bisector of OB with the y-axis. Let D = (0, x).

Triangle OAC yields sinψ = OAOC

=1

2

√x2+(ax2)2

r, triangle OBD yields sinψ =

BDOB

= ax2

√x2+a2x4

, and1

2

√x2+a2x4

r= ax2

√x2+a2x4

, so that 12 (x

2 + a2x4) = arx2,

and 12 (1 + a2x2) = ar, so that r = 1+a2x2

2a . Taking the limit as x→ 0, we ob-

tain r = 12a , hence k = 1

r= 2a = f ′′(0). Thus the curvature of the parabola at

its vertex equals the second derivative with respect to x (even though x is not thearclength parameter of the graph).


Example 4.5.2. If C is a circle, then the intersection point of twonormals is always the center of C even if the normals are not close toeach other. Thus the center and the center of curvature q coincide. Fora general curves we need to take the normals close to each other.

Theorem 4.5.3 (Relation between center of curvature and oscu-lating circle). At a point p ∈ C, consider the osculating circle to thecurve. Then the corresponding center of curvature q of C (the intersec-tion point of two infinitely close normals) is the center of the osculatingcircle.

Definition 4.5.4. Let p ∈ C. The radius of curvature R of acurve C at p is the distance from p to the center of curvature q.Namely, R = |pq|.

Corollary 4.5.5. The curvature kα at p is the inverse of the radiusof curvature: kα = 1

R= 1

|pq| .

4.6. Flat Laplacian

Computing the curvature of implicitly defined curves C ⊆ R2 willinvolve a certain second-order differential operator. We start with thesimplest example of a second-order operator, the Laplacian.

Definition 4.6.1. The flat Laplacian ∆0 on R2 is the differentialoperator is defined in the following two equivalent ways:

(1) ∆0 =∂2

∂x2+ ∂2

∂y2;

(2) we apply ∆0 to a smooth function F = F (x, y) by setting

∆0F = ∂2F∂x2

+ ∂2F∂y2

.

We also introduce the traditional briefer notation

Fxx =∂2F

∂x2, Fyy =

∂2F

∂y2, Fxy =

∂2F

∂x ∂y.

Then we obtain the equivalent formula

∆0(F ) = Fxx + Fyy.

Example 4.6.2. If F (x, y) = x2 + y2 − r2, then ∂2F∂x2

= 2 and

similarly ∂2F∂y2

= 2, hence ∆0F = 4.

4.7. Bateman–Reiss operator

Recall that we have two types of presentations of a curve, eitherparametric or implicit.

4.8. GEODESIC CURVATURE FOR AN IMPLICIT CURVE 53

Remark 4.7.1. In Section 4.2 the curvature of a curve was calcu-lated starting with a parametric representation of the curve. If a curveis given implicitly as the locus (solution set) of an equation F (x, y) = 0,one can also calculate the geodesic curvature, by means of the theoremsgiven in the Section 4.8, and exploiting the Bateman–Reiss operator ofDefinition 4.7.2.

We will be interested in the following second-order operator thatappears in a formula for curvature.

Definition 4.7.2. The Bateman–Reiss operator DB is defined by

DB(F ) = FxxF2y − 2FxyFxFy + FyyF

2x . (4.7.1)

Remark 4.7.3. DB is a non-linear second order differential opera-tor.

The subscript “B” stands for Bateman, as in the Bateman equa-tion FxxF

2y − 2FxyFxFy + FyyF

2x = 0.

Theorem 4.7.4. The Bateman operator can be represented by thedeterminant

DB(F ) = −det

0 Fx FyFx Fxx FxyFy Fxy Fyy

. (4.7.2)

Proof. Expanding the determinant (4.7.2) along the first row, weobtain the formula (4.7.1). �

Remark 4.7.5 (Historical remark). This operator was treated indetail in [Goldman 2005, p. 637, formula (3.1)]. The same operatoroccurs in the Reiss3 relation in algebraic geometry; see Griffiths andHarris [GriH78, p. 677]. See also Valenti [4, p. 804] who refers togeodesic curvature as isophote curvature in the context of a study ofluminosity and eye center location. See also [5] and (two) referencestherein.

4.8. Geodesic curvature for an implicit curve

The operator DB defined in Section 4.7 allows us to calculate thecurvature of a curve presented in implicit form, without having to spec-ify a parametrisation. Let CF ⊆ R2 be a curve defined implicitlyby F (x, y) = 0.

3Michel Reiss (1805-1869).


Theorem 4.8.1. Let p ∈ CF , and suppose ∇F (p) 6= 0. Then thegeodesic curvature kC of CF at the point p is given by

kC =|DB(F )||∇F |3 ,

where DB is the Bateman–Reiss operator defined in (4.7.1).

A proof can be found in [Goldman 2005].

4.9. Curvature of circle via DB

We will use the formula for curvature based on the Bateman–Reissoperator to calculate the curvature and solve maximization problems.We start with the circle.

Example 4.9.1. The circle C of radius r is defined by the equa-tion F (x, y) = 0, where F = x2 + y2 − r2. Then

Fx = 2x, Fy = 2y, ∇F = (2x, 2y)t,

and therefore |∇F | = 2√x2 + y2 = 2r. Meanwhile, Fxx = 2, Fyy =

2, Fxy = 0, hence

DB(F ) = 2(2y)2 + 2(2x)2 = 8r2,

and therefore the curvature is kC = 8r2

8r3= 1

r, in agreement with Theo-

rem 4.2.4.

4.10. Curvature of parabola via DB

Consider the parabola C = CF given by the zero locus of F (x, y) =y − x2. Then

Fx = −2x, Fy = 1, |∇F | =√1 + 4x2.

Meanwhile, Fxx = −2, Fyy = 0, Fxy = 0. Applying the formulaDB(F ) =FxxF

2y − 2FxyFxFy + FyyF

2x we obtain DB(F ) = −2(1)2 = −2. We ob-

tain the following theorem.

Theorem 4.10.1. The curvature of the parabola y = x2 satisfies

kC =2

(1 + x2)3/2. (4.10.1)

This enables us to identify the point of maximal curvature in The-orem 4.10.2 as follows.

Theorem 4.10.2. The apex of the parabola y = x2 is its point ofmaximal curvature.

4.11. CURVATURE OF HYPERBOLA VIA DB 55

Proof. By formula (4.10.1), kC = 2(1+x2)3/2

. Since 1 + x2 ≥ 1, the

curvature attains its maximal value k = 2 when x = 0. �

Corollary 4.10.3. The circle of curvature of the parabola at theorigin has radius R = 1

2.

4.11. Curvature of hyperbola via DB

Consider the hyperbola C = CF given by the zero locus of F (x, y) =xy − 1. Then

Fx = y, Fy = x, |∇F | =√x2 + y2.

Meanwhile Fxx = 0, Fyy = 0, and Fxy = 1. Hence DB(F ) = −2xy. Tosimplify the expression for DB(F ) we exploit the defining equation ofthe curve xy = 1. Hence we have DB(F ) = −2 at every point of thehyperbola. We obtain the following theorem.

Theorem 4.11.1. The curvature of the hyperbola y = 1xis

kC =2

(x2 + y2)3/2. (4.11.1)

This enables us to find the maximum in Theorem 4.11.2 as follows.

Theorem 4.11.2. The curvature of the hyperbola C defined by theequation xy = 1 is maximal at the points (1, 1) and (−1,−1).

Proof. Formula (4.11.1) gives kC = 2(x2+y2)3/2

.

Step 1. In order to maximize this expression along the curve, itsuffices to minimize the expression x2+ y2 whose power appears in thedenominator, restricted to the curve itself. We can assume without lossof generality that x > 0. We exploit the defining relation xy = 1 of thecurve to obtain

x2 + y2 = (x+ y)2 − 2xy = (x+ y)2 − 2 =(x+ 1

x

)2 − 2.

Step 2. To minimize this expression, it suffices to minimize thequantity x+ 1

x. We will show that the sum x+ 1

xis minimal when x = 1.

Namely, to show that x+ 1x≥ 2, rewrite the inequality as x2 + 1 ≥ 2x

or x2+1−2x ≥ 0 or equivalently (x−1)2 ≥ 0 which is a true inequality.Hence the maximum of the curvature is when x = 1 and so y = 1,proving the theorem.

The same result can be obtained by differentiating the expression(x+ 1

x

)2 − 1. �


4.12. Maximal curvature of a logarithmic curve

Theorem 4.12.1. Consider the logarithmic curve

C = {(x, y) : y = ln x, x > 0}.Then the maximum of curvature of the curve is attained at the pointx = 1√

2, y = − ln 2

2.

Proof. Let F (x, y) = y − ln x. Then

Fx = −1

x, Fy = 1, |∇F | =

√1 + x−2.

Then Fxx =1x2, Fyy = 0, and Fxy = 0. Applying the formula DB(F ) =

FxxF2y−2FxyFxFy+FyyF

2x we obtainDB(F ) =

1x2·1−2·0+0 = 1

x2= x−2.

The curvature of the logarithmic curve is therefore

kC =x−2

(√1 + x−2)3

=1

x2(1 + x−2)3/2=

x3

x2(x2 + 1)3/2=

x

(x2 + 1)3/2.

To maximize kC it is sufficient to maximize the expression k2C = x2

(x2+1)3.

We will use an auxiliary variable z = x2 to simplify calculations.We are therefore interested in maximizing the function

g(z) =z

(z + 1)3.

Differentiating we obtain

g′(z) =(z + 1)3 · 1− z · 3(z + 1)2

(z + 1)6

=z + 1− 3z

(z + 1)4=

1− 2z

(z + 1)4= 0.

We obtain an extremum when 1 − 2z = 0, i.e., z = 12. Checking

that the second derivative is negative at the point, we conclude thatthis is a point of maximum. Thus the maximum of the curvatureof the logarithmic curve is attained when x2 = 1

2, i.e., x = 2−1/2.

Hence y = ln(2−1/2) = −12ln 2. �

4.13. Exploiting the defining equation in studying curvature

The defining equation of the curve may be exploited several timesin the course of the calculation when studying the curvature of a curve.Let us calculate out a specific example as follows.

4.13. EXPLOITING THE DEFINING EQUATION IN STUDYING CURVATURE57

Theorem 4.13.1. The maxima of the curvature of the curve

C = {(x, y) : xy + y2 − 1 = 0} (4.13.1)

are at the two points with coordinates

x = ±√2− 14√2

, y = ± 14√2.

Proof. Set F (x, y) = xy + y2 − 1. Then

Fx = y, Fy = x+2y, |∇F | =√y2 + (x+ 2y)2 =

√y2 + x2 + 4xy + 4y2

and therefore

|∇F | =√y2 + x2 + 4(xy + y2). (4.13.2)

We will give a proof in four steps.

Step 1. We use the defining equation of the curve to replace xy+y2

by 1. This enables us to simplify (4.13.2) to

|∇F | =√x2 + y2 + 4. (4.13.3)

Next, Fxx = 0, Fxy = 1, and Fyy = 2. Therefore

DB(F ) = 0− 2xy − 4y2 + 2y2 = −2(xy + y2). (4.13.4)

Step 2. Exploiting the defining relation of the curve, we obtainfrom equation (4.13.4) that

DB(F ) = −2. (4.13.5)

Using (4.13.3) and (4.13.5), we obtain that the curvature at a point (x, y)of the curve is

kC =|DB(F )||∇F |3 =

2

(x2 + y2 + 4)3/2. (4.13.6)

The curvature (4.13.6) of the curve is maximal when the expression x2+y2+4 along the curve is minimal, or equivalently when x2+y2 is minimalalong the curve.

Step 3. Using the defining relation of the curve we obtain xy =1− y2 or

x =1− y2

y=

1

y− y. (4.13.7)

Therefore

x2 + y2 =(1− y2)2

y2+ y2 =

(1− y2)2 + y4

y2.


To simplify calculations, we introduce an auxiliary variable z = y2.Thus we have

x2 + y2 =(1− z)2 + z2

z(4.13.8)

Step 4. The expression (4.13.8) to be minimized is

g(z) =(1− z)2 + z2

z=

2z2 − 2z + 1

z= 2z +

1

z− 2.

We set g′(z) = 2− 1z2

= 0 to find the extremum z = 1√2. Checking that

the second derivative is positive, we conclude that y = ±√z = ± 1

21/4.

The corresponding x can be found from x = 1−y2y

. This gives the

maximum of the curvature for the curve (4.13.1). �

4.14. Curvature of graph of function

Theorem 4.14.1. Let c be a critical point of f(x). Consider thegraph of f at the point (c, f(c)) ∈ R2. Then the curvature k of thegraph equals

k = |f ′′(c)|.Proof. We parametrize the graph by α(t) = (t, f(t)). Then we

have ∂2α∂t2

= (0, d2fdt2

). Therefore at the critical point c, we have∣∣∣∂2α∂t2 (c)

∣∣∣ =∣∣∣d2fdt2 (c)∣∣∣. This is the expected answer at a critical point. However, this

argument is merely a heuristic calculation because the parametrisa-tion (t, f(t)) is not a unit speed parametrisation of the graph.

We apply the characterisation of curvature in terms of the BatemanoperatorDB(F ) = FxxF

2y−2FxyFxFy+FyyF

2x . Let F (x, y) = −f(x)+y.

At the critical point c, we have

∇F = (−f ′(c), 1)t = (0, 1)t,

whileDB(F ) = −f ′′(x)(1)2 − 0 + 0 = −f ′′(x).

Hence the curvature of the graph at this point satisfies

k =|DBF ||∇F |3 =

|f ′′(c)|1

= |f ′′(c)|,

as required. �

Example 4.14.2. At the apex of the parabola y = x2 the derivativeis 2 and therefore the curvature is 2, while the radius of the osculatingcircle is 1

2, which agrees with Corollary 4.10.3.

CHAPTER 5

Surfaces and their curvature

5.1. Existence of arclength

Consider a geometric curve C ⊆ R2. Assume C admits a regu-lar parametrisation. We will now prove the existence of a unit speedparameter s, i.e., the arclength parameter.

Theorem 5.1.1. Let α(t) be a regular parametrisation of the geo-metric curve C ⊆ R2. Then there exists a unit speed parametritionβ(s) = α(t(s)) of the curve C, where the parameter s is defined by

s(t) =

∫ t

a

∣∣∣dαdτ

∣∣∣dτ .

Proof. Recall the formula for the length of the plane curve definedby the graph of a function f(x) from a to b:

L =

∫ b

a

√1 + f ′(x)2 dx .

More generally, by the Pythagorean theorem applied to a curve α(t) =(α1(t), α2(t)), we have the following formula for the length:

L =

∫ b

a

√(dα1

dt

)2+(dα2

dt

)2dt

=

∫ b

a

∣∣∣dαdt

∣∣∣dt.

We define the new parameter s = s(t) by setting

s(t) =

∫ t

a

∣∣∣dαdτ

∣∣∣dτ, (5.1.1)

where τ is a dummy variable (internal variable of integration). TheFundamental Theorem of Calculus applied to (5.1.1) yields

ds

dt=∣∣∣dαdt

∣∣∣.59

60 5. SURFACES AND THEIR CURVATURE

The function being monotone increasing, there exists an inverse func-tion t = t(s). Let β(s) = α(t(s)). Then by chain rule

dβ

ds=dα

dt

dt

ds=dα

dt

1

ds/dt=dα

dt

1

|dα/dt| .

Thus∣∣dβds

∣∣ = 1. �

We provide an example of non-existence of a regular parametrisa-tion for a curve.

Example 5.1.2 (Cusp). The plane curve α(t) = (t3, t2) is smoothbut not regular. Its graph exhibits a cusp.1 In this case it is im-possible to find a smooth arclength parametrisation of the curve. InTheorem 5.12.4 it is shown that its curvature tends to infinity as itapproaches the cusp.

Remark 5.1.3 (Curves in R3). A space curve may be written incoordinates as

α(s) = (α1(s), α2(s), α3(s)).

Here s is the arc length if∣∣∣dαds∣∣∣ = 1 i.e.

∑3i=1

(dαi

ds

)2= 1.

Example 5.1.4. Helix α(t) = (a cosωt, a sinωt, bt).(i) make a drawing in case a = b = ω = 1.(ii) parametrize by arc length.(iii) compute the curvature.

5.2. Surfaces; Arnold’s observation on folding a page

The differential geometry of surfaces in Euclidean 3-space startswith the observation that they inherit a metric structure from the am-bient space (i.e. the Euclidean space).

Question 5.2.1. We would like to understand which geometricproperties of this structure are intrinsic?2

Part of the job is to clarify the sense of the term intrinsic.

Remark 5.2.2. Following Arnold [Ar74, Appendix 1, p. 301], notethat a piece of paper may be placed flat on a table, or it may be rolledinto a cylinder, or it may be rolled into a cone. In mathematical termsthis can be formulated by saying that

the plane, the cylinder, and the cone (apart from thevertex) have the same local intrinsic geometry.

1chod2This is atzmit rather than pnimit according to Vishne.

5.3. REGULAR SURFACE; JACOBIAN 61

However, the piece of paper cannot be transformed into the surface ofa sphere, that is, without tearing or stretching. Understanding thisphenomenon quantitatively is our goal, cf. Figure 11.11.1.

5.3. Regular surface; Jacobian

Consider a surface M ⊆ R3 parametrized by a map x(u1, u2) or

x : R2 → R3. (5.3.1)

We will always assume that x is differentiable.

Definition 5.3.1. The Jacobian matrix Jx is the 3× 2 matrix

Jx =

(∂xi

∂uj

), (5.3.2)

where xi, i = 1, 2, 3, are the three components of the vector valuedfunction x.

Definition 5.3.2. The parametrisation x of the surfaceM is calledregular if one of the following equivalent conditions is satisfied:

(1) the vectors ∂x∂u1

and ∂x∂u2

are linearly independent;

(2) the vector product x1 × x2 is nonzero, where xi =∂x∂ui

;(3) the Jacobian matrix (5.3.2) is of rank 2.

Example 5.3.3 (Regular parametrisation of hemisphere). Let b > 0

be a fixed real number, and let f(x, y) =√b2 − x2 − y2. The graph of

the function f in R3 can be parametrized as follows:

x+(u1, u2) = (u1, u2, f(u1, u2)). (5.3.3)

This provides a parametrisation of the (open) northern hemisphere.

Then ∂x+

∂u1= (1, 0, fx(u

1, u2))t while ∂x+

∂u2= (0, 1, fy(u

1, u2))t. These vec-tors are linearly independent and therefore (5.3.3) is a regular parametri-sation.

Note that we have the relations

fx =−xf

=−x√

b2 − x2 − y2, fy =

−yf. (5.3.4)

The southern hemisphere can be similarly parametrized by

x− = (u1, u2,−f(u1, u2)).The formulas for the ∂x−

∂uiare then ∂x−

∂u1= (1, 0,−fx)t = (1, 0, x√

b2−x2−y2)t,

while ∂x−

∂u2= (0, 1,−fy)t = (0, 1, y√

b2−x2−y2)t.


5.4. Coefficients of first fundamental form of a surface

Our starting point is the first fundamental form of a surface, ob-tained by restricting the 3-dimensional inner product.

Remark 5.4.1. What does the first fundamental form measure? Ahelpful observation to keep in mind is that it enables one to measurethe length of curves on the surface.

Let 〈 , 〉 denote the inner product in R3. For i = 1, 2 and j = 1, 2,define functions gij = gij(u

1, u2) called “metric coefficients” by

gij(u1, u2) =

⟨ ∂x∂ui

,∂x

∂uj

⟩. (5.4.1)

Remark 5.4.2. We have gij = gji as the inner product is symmet-ric.

Example 5.4.3 (Metric coefficients for the graph of a function).The surface defined by the graph of a function f = f(x, y) as in Ex-ample 5.3.3 satisfies

⟨∂x

∂u1,∂x

∂u1

⟩= 1 + f 2

x ,

⟨∂x

∂u1,∂x

∂u2

⟩= fxfy,

⟨∂x

∂u2,∂x

∂u2

⟩= 1 + f 2

y .

Therefore in this case we have the matrix of metric coefficients

(gij) =

(1 + f 2

x fxfyfxfy 1 + f 2

y

)

In the case of the hemisphere, the partial derivatives are as in for-mula (5.3.4).

5.5. Metric coefficients in spherical coordinates

Spherical coordinates3 (ρ, θ, ϕ) in 3-space will be reviewed in moredetail in Section 6.7. Briefly, ϕ is the angle formed by the positionvector of the point with the positive direction of the z-axis. Mean-while, θ is the polar coordinate angle θ for the projection of the vectorto the x, y plane.

3Koordinatot kaduriot

5.6. TANGENT PLANE; GRAM MATRIX 63

Example 5.5.1 (Metric coefficients in spherical coordinates). Con-sider the parametrisation of the unit sphere S2 by means of sphericalcoordinates, so that x = x(θ, ϕ). We have the Cartesian coordinates

x = sinϕ cos θ, y = sinϕ sin θ, z = cosϕ .

Setting u1 = θ and u2 = ϕ, we obtain

∂x

∂u1= (− sinϕ sin θ, sinϕ cos θ, 0)

and∂x

∂u2= (cosϕ cos θ, cosϕ sin θ,− sinϕ),

Thus in this case we have

(gij) =

(sin2 ϕ 00 1

)

so that det(gij) = sin2 ϕ and√det(gij) = sinϕ. This expression will

appear in the formula for the area in Section 8.3.

5.6. Tangent plane; Gram matrix

Consider a parametrisation x(u1, u2) of a surface M ⊆ R3. Let p =x(u1, u2) ∈ R3 be a particular point on the surface.

Definition 5.6.1. The vectors xi =∂x∂ui

, i = 1, 2, are called thetangent vectors to the surface M at the point p = x(u1, u2).

Definition 5.6.2. Given an ordered n-tuple S = (vi)i=1,...,n in Rb,we define its Gram matrix as the matrix of inner products

Gram(S) = (〈vi, vj〉)i=1,...,n; j=1,...,n. (5.6.1)

In Section 5.5 we defined the metric coefficients gij = gij(u1, u2).

We now state a relationship between the Gram matrix and the metriccoefficients.

Theorem 5.6.3. The 2× 2 matrix (gij) is the Gram matrix of thepair of tangent vectors:

(gij) = JTx Jx,

cf. formula (14.6.1).

The proof is immediate.

Definition 5.6.4. The plane in R3 spanned by the vectors x1(u1, u2)

and x2(u1, u2) is called the tangent plane to the surface M at the

point p = x(u1, u2), and denoted

TpM.


5.7. First fundamental form Ip of M

The tangent plane TpM of a surface M ⊆ R3 was defined in Sec-tion 5.6.

Definition 5.7.1. The first fundamental form Ip of the surface Mat the point p is the bilinear form on the tangent plane Tp, namely

Ip : TpM × TpM → R,

defined by the restriction of the ambient Euclidean inner product:

Ip(v, w) = 〈v, w〉R3 ,

for all v, w ∈ TpM . With respect to the basis (frame) (x1, x2), the firstfundamental form is given by the matrix (gij), where

gij = 〈xi, xj〉.The coefficients gij are sometimes called ‘metric coefficients.’

5.8. The form Ip of plane and cylinder

Like curves, surfaces can be represented either implicitly or para-metrically (see Section 3.11 for representations of curves).

The example of the sphere was discussed in Section 5.5. We nowconsider two more examples.

Example 5.8.1 (Plane). The x, y-plane in R3 is defined implic-itly by the equation z = 0. Consider the parametrisation x(u1, u2) =(u1, u2, 0) ∈ R3. This is a parametrisation of the xy-plane in R3.Then x1 = (1, 0, 0)t, x2 = (0, 1, 0)t, and

g11 = 〈x1, x1〉 =⟨100

,

100

⟩

= 1,

etcetera. Thus we have (gij) =

(1 00 1

)= I.

Example 5.8.2 (Cylinder). The cylinder in R3 is defined implicitlyby the equation x2 + y2 = 1. Let x(u1, u2) = (cos u1, sin u1, u2). Thisformula provides a parametrisation of the cylinder. We have

x1 = (− sin u1, cos u1, 0)t

and x2 = (0, 0, 1)t, while

g11 =

⟨− sin u1

cos u1

0

,

− sin u1

cos u1

0

⟩

= sin2 u1 + cos2 u1 = 1,

5.9. SURFACES OF REVOLUTION 65

etc. Thus (gij) =

(1 00 1

)= I.

Remark 5.8.3. The two examples above illustrate that the firstfundamental form does not contain all the information (even up toorthogonal transformations) about how the surface sits in R3. Indeed,the plane and the cylinder have the same first fundamental form, butare geometrically distinct embedded surfaces.

5.9. Surfaces of revolution

For surfaces of revolution, it is customary to use the notation u1 = θand u2 = φ. The starting point is a generating curve C in the xz-plane,parametrized by a pair of functions

x = r(φ), z = z(φ).

Definition 5.9.1. The surface of revolution (around the z-axis)generated by C is parametrized as follows:

x(θ, φ) = (r(φ) cos θ, r(φ) sin θ, z(φ)). (5.9.1)

Example 5.9.2. Consider a generating curve which is the verticalline r(φ) = 1, z(φ) = φ. The resulting surface of revolution is thecylinder.

Example 5.9.3. The generating curve r(φ) = sinφ, z(φ) = cosφyields the sphere S2 in spherical coordinates as discussed in Section 5.5;see Example 5.9.5 for more details.

Theorem 5.9.4. Assume that φ is the arclength parameter of aparametrisation (r(φ), z(φ)) of the generating curve C. Then the firstfundamental form of the corresponding surface of revolution (5.9.1) isgiven by

(gij) =

(r2(φ) 00 1

).

Proof. We have

x1 =∂x

∂θ= (−r sin θ, r cos θ, 0)t,

x2 =∂x

∂φ=

(dr

dφcos θ,

dr

dφsin θ,

dz

dφ

)t

therefore

g11 =

∣∣∣∣∣∣

−r sin θr cos θ

0

∣∣∣∣∣∣

2

= r2 sin2 θ + r2 cos2 θ = r2


and

g22 =

∣∣∣∣∣∣

drdφ

cos θdrdφ

sin θdzdφ

∣∣∣∣∣∣

2

=

(dr

dφ

)2

(cos2 θ + sin2 θ) +

(dz

dφ

)2

=

(dr

dφ

)2

+

(dz

dφ

)2

and

g12 =

⟨−r sin θr cos θ

0

,

drdφ

cos θdrdφ

sin θdzdφ

⟩

= −r drdφ

sin θ cos θ + rdr

dφcos θ sin θ = 0.

Thus

(gij) =

(r2 0

0(drdφ

)2+(dzdφ

)2).

In the case of an arclength parametrisation of the generating curve C,we obtain g22 = 1, proving the theorem. �

Example 5.9.5. Consider the curve (sinφ, cosφ) in the x, z-plane.The resulting surface of revolution is the sphere, where the φ parame-ter coincides with the angle ϕ of spherical coordinates. Thus, for thesphere S2 we obtain

(gij) =

(sin2 φ 00 1

).

5.10. From tractrix to pseudosphere

Definition 5.10.1. The tractrix is the curve is parametrized by(r(φ), z(φ)) where r(φ) = eφ and

z(φ) =

∫ φ

0

√1− e2ψdψ = −

∫ 0

φ

√1− e2ψdψ,

where −∞ < φ ≤ 0.

Example 5.10.2. The pseudosphere (so called because its Gauss-ian curvature equals −1) is the surface of revolution generated by thetractrix. Here g11 = e2φ, while

g22 = (eφ)2 +(√

1− e2φ)2

= e2φ + 1− e2φ = 1.

5.12. MEASURING LENGTH OF CURVES ON SURFACES 67

Thus (gij) =

(e2φ 00 1

).

The Gaussian curvature of the pseudosphere will be calculated inSection 13.3.

5.11. Chain rule in two variables

How does one measure the length of curves on a surface in terms ofthe metric coefficients of the surface x(u1, u2)? Let

α : [a, b] → R2, α(t) =

(α1(t)α2(t)

).

be a plane curve. We will exploit the Einstein summation convention.We will also use the following version of chain rule in several variables.

Theorem 5.11.1 (Chain rule in two variables). Consider the curveon the surface defined by the composition β(t) = x ◦ α(t) where x =

x(u1, u2). Then dβdt

=∑2

i=1∂x∂ui

dαi

dt, or in Einstein summation conven-

tion

dβ

dt=

∂x

∂uidαi

dt= Jx

dα

dt.

For a proof see Keisler [Ke74, p. 672].

5.12. Measuring length of curves on surfaces

In Section 5.5, we defined the metric coefficients gij = 〈xi, xj〉 of asurface M with parametrisation x(u1, u2). The first fundamental form

Ip : TpM × TpM → R,

represented by the matrix (gij), determines the length of curves on thesurface as follows.

Theorem 5.12.1. Consider a surface with parametrisation x(u1, u2).Let β(t) = x ◦ α(t), t ∈ [a, b], be a curve on the surface. Then thelength L of β is given by the formula

L =

∫ b

a

√gij(α(t)

)dαidt

dαj

dtdt.


Proof. The length L of β is calculated as follows using the chainrule:

L =

∫ b

a

∣∣∣∣dβ

dt

∣∣∣∣ dt

=

∫ b

a

∣∣∣∣∣

2∑

i=1

∂x

∂uidαi

dt

∣∣∣∣∣ dt

=

∫ b

a

√⟨xidαi

dt, xj

dαj

dt

⟩dt

=

∫ b

a

√〈xi, xj〉

dαi

dt

dαj

dtdt

=

∫ b

a

√gij(α(t))

dαi

dt

dαj

dtdt.

We therefore obtain L =∫ ba

√gij(α(t))

dαi

dtdαj

dtdt as required. �

Corollary 5.12.2. If (gij) = (δij) is the identity matrix, then thelength of the curve β = x ◦ α on the surface equals the length of theoriginal curve α in R2.

Proof. If the first fundamental form of the surface satisfies gij =

δij, then L =∫ ba

√(dα1

dt

)2+(dα2

dt

)2dt, namely the length of the curve α(t)

in R2. �

Remark 5.12.3. To simplify notation, let x = x(t) = α1(t) andy = y(t) = α2(t). Then the dt cancels out. The infinitesimal element

of arclength is ds =√dx2 + dy2. The length of the curve is

∫ds =∫ √

dx2 + dy2.

Theorem 5.12.4. The curvature of the cusp curve α(t) = (t3, t2)of Example 5.1.2 tends to infinity as we approach the cusp.

Proof. The curve can be represented implicitly as the level curveof the function F (x, y) = x2− y3. We have Fx = 2x, Fy = 3y2, |∇F | =√4x2 + 9y4. Meanwhile Fxx = 2, Fyy = 6y, Fxy = 0. Thus DBF =

2(9y4) + 6y(4x2) = 18y4 + 24x2y = 6y(3y3 + 4x2). From the definingequation of the curve we obtain DBF = 6y(3y3 − 4y3) = −6y4. Thusthe curvature of the curve is

kC =6y4

(4y3 + 9y4)3/2=

6y4

y9/2(4 + 9y)3/2=

6

y1/2(4 + 9y)3/2.

We see in particular that as we approach the cusp (i.e., as y tends tozero), the curvature tends to infinity. �

CHAPTER 6

Gamma symbols of a surface

In this chapter we start studying the intrinsic geometry of surfaces.

6.1. Normal vector to a surface

We first review the notation introduced in Section 5.3.

• x : R2 → R3 is a regular parametrized surface;• xi = ∂x

∂ui, i = 1, 2 are its tangent vectors;

• the tangent vectors span the tangent plane Tp = Span(x1, x2)at a point p = x(u1, u2) of the surface.

Definition 6.1.1. The normal vector n(u1, u2) to a regular surfaceat the point x(u1, u2) ∈ R3 is defined in terms of the vector product,cf. Definition 1.10.1, as follows:

n =x1 × x2|x1 × x2|

,

so that 〈n, xi〉 = 0 for each i = 1, 2.

Remark 6.1.2 (Sign ambiguity). Either the vector x1×x2|x1×x2| or the

opposite vectorx2 × x1|x1 × x2|

= − x1 × x2|x1 × x2|

can be taken to be a normal vector to the surface.

Theorem 6.1.3. Consider the surface M ⊆ R3 given by the graphz = f(x, y) of a function f(x, y). Then the vector

(fx, fy,−1)t

is orthogonal to the tangent plane Tp toM at the point p = (x, y, f(x, y)).

Proof. A standard parametrisation of the graph of f is

x(u1, u2) = (u1, u2, f(u1, u2)).

Then x1 = (1, 0, fx)t and x2 = (0, 1, fy)

t. Taking the cross product, weobtain

det

−→i

−→j

−→k

1 0 fx0 1 fy

= −fx

−→i − fy

−→j +

−→k

69

70 6. GAMMA SYMBOLS OF A SURFACE

which is the opposite of the vector (fx, fy,−1)t. Note that this maynot be a unit vector. �

Corollary 6.1.4. The normal vector n of the graph of f(x, y)in R3 is

n =(fx, fy,−1)t√f 2x + f 2

y + 1

up to sign.

Proof. This is immediate from the theorem by normalizing thevector (fx, fy,−1)t. �

6.2. Γ symbols of a surface

The symbols Γkij, roughly speaking,1 account for how the surface

twists2 in space. They are, however, coordinate dependent (unlike theGaussian curvature that we will define in Section 9.7).

Remark 6.2.1. We will see that the symbols Γkij also control thebehavior of geodesics on the surface. Here geodesics on a surface canbe thought of as curves that are to the surface what straight lines areto a plane, or what great circles are to a sphere; cf. Definition 7.9.2.

Remark 6.2.2. The metric coefficients gij were defined in terms offirst partial derivatives of x. Meanwhile, the symbols Γkij are defined

in terms of the second partial derivatives.3

Lemma 6.2.3. If a parametrisation x is regular then the vectors(x1, x2, n) form a basis (frame) for R3.

The coefficients Γ are defined as the coefficients of the decomposi-tion of the second partial derivative vector xij , defined by

xij =∂2x

∂ui∂uj

with respect to the basis, or frame,4 (x1, x2, n), as follows.

Definition 6.2.4. The symbols Γkij are uniquely determined by theformula

xij = Γ1ij x1 + Γ2

ij x2 + Lijn

Note that the coefficients Lij are also uniquely defined by the for-mula. They will be analyzed in Section 9.8.

1In the first approximation: kiruv rishon.2Mitpatelet3Since they are not even tensors, one does not stagger the indices.4Maarechet yichus

6.3. BASIC PROPERTIES OF THE Γ SYMBOLS 71

6.3. Basic properties of the Γ symbols

Proposition 6.3.1. We have the following formula for the Gammasymbols:

Γkij = 〈xij , xℓ〉gℓk, (6.3.1)

where (gℓk) is the inverse matrix of (gij).

Proof. We will drop the underlines for the purposes of the proof.We have

〈xij, xℓ〉 = 〈Γkijxk + Lijn, xℓ〉= 〈Γkijxk, xℓ〉+ 〈Lijn, xℓ〉= Γkijgkℓ

since the vector n is perpendicular to each tangent vector of the surface.We now multiply by gℓm and sum over the index ℓ:

〈xij, xℓ〉gℓm = Γkijgkℓgℓm = Γkijδ

mk = Γmij .

This is equivalent to the desired formula. �

Remark 6.3.2. We have the following relation: Γkij = Γkji, or Γk[ij] =

0 for all i, j, k.

Theorem 6.3.3. For the standard parametrisations of the planeand the cylinder, the Gamma symbols vanish.

Proof. We calculate the symbols Γkij for the plane x(u1, u2) =

(u1, u2, 0). We have x1 = (1, 0, 0)t, x2 = (0, 1, 0)t. Thus we have xij = 0for all i, j. Hence by formula (6.3.1), Γkij = 0 for all i, j, k.

We calculate the symbols for the cylinder x(u1, u2) = (cos u1, sin u1, u2).We obtain the following data:

x1 = (− sin u1, cos u1, 0)t;

x2 = (0, 0, 1)t;

the normal vector n = (cos u1, sin u1, 0)t.

Thus we have x22 = 0, x21 = 0 and so Γk22 = 0 and Γk12 = Γk21 = 0 forall k.

Meanwhile, x11 = (− cos u1,− sin u1, 0)t. This vector is propor-tional to n:

x11 = 0x1 + 0x2 + (−1)n.

Hence Γk11 = 0 for all k. �

An example of a surface with nonzero Γ symbols will appear inSection 6.6.


6.4. Derivatives of the metric coefficients

Definition 6.4.1. We will use the following comma notation forthe partial derivative of the function gij = gij(u

1, u2):

gij,k =∂

∂uk(gij).

Lemma 6.4.2. Scalar product of vector valued functions f, g satisfiesLeibniz’s rule:

〈f, g〉′ = 〈f ′, g〉+ 〈f, g′〉. (6.4.1)

Proof. See [Leib]. In more detail, let (f1, f2) be components of f ,and let (g1, g2) be components of g. Then

〈f, g〉′ = (f1g1 + f2g2)′ = f1g

′1 + f ′

1g1 + f2g′2 + f ′

2g2 = 〈f ′, g〉+ 〈f, g′〉.The same proof goes through for arbitrary number of components, e.g.,for functions with values in R3. �

Proposition 6.4.3. In terms of the symmetrisation notation in-troduced in Section 1.6, we have

gij,k = 2gm{iΓmj}k, (6.4.2)

or more explicitlygij,k = gmiΓ

mjk + gmjΓ

mik. (6.4.3)

Proof. Indeed, by Leibniz rule (Lemma 6.4.2),

gij,k =∂

∂uk〈xi, xj〉

= 〈xik, xj〉+ 〈xi, xjk〉= 〈Γmikxm, xj〉+ 〈Γmjkxm, xi〉= Γmik〈xm, xj〉+ Γmjk〈xm, xi〉.

The scalar products above by definition are the metric coefficients.Therefore we have

gij,k = Γmikgmj + Γmjkgmi

= gmjΓmik + gmiΓ

mjk

= 2gm{jΓmi}k

= 2gm{iΓmj}k

as required. �

Remark 6.4.4. The system of equations of type (6.4.3) suggeststhat one may be able to solve the system for Γkij, i.e., to express Γkijin terms of the metric coefficients gij and their derivatives gij,k. Thisturns out to be correct, as we show in Section 6.5.

6.6. GAMMA SYMBOLS FOR A SURFACE OF REVOLUTION 73

6.5. Intrinsic nature of the Γ symbols

The coefficients Γ are intrinsic5 meaning that they are determinedby the metric coefficients alone, and are therefore independent of theambient (extrinsic) geometry of the surface, i.e., the way the surface“sits” in 3-space.

Theorem 6.5.1. The symbols Γkij can be expressed in terms of thefirst fundamental form and its derivatives as follows:

Γkij =1

2

(giℓ,j − gij,ℓ + gjℓ,i

)gℓk,

where gij is the inverse matrix of gij.

Proof. The proof is a calculation. Applying Proposition 6.4.3three times, we obtain

giℓ,j − gij,ℓ + gjℓ,i = 2gm{iΓmℓ}j − 2gm{iΓ

mj}ℓ + 2gm{jΓ

mℓ}i

= gmiΓmℓj + gmℓΓ

mij − gmiΓ

mjℓ − gmjΓ

miℓ + gmjΓ

mℓi + gmℓΓ

mji

= 2gmℓΓmji .

Thus1

2(giℓ,j − gij,ℓ + gjℓ,i)g

ℓk = Γmij gmℓgℓk = Γmij δ

km = Γkij, as required.

6

�

6.6. Gamma symbols for a surface of revolution

Recall that a surface of revolution is obtained by starting with acurve (r(φ), z(φ)) in the (x, z) plane, and rotating it around the z-axis,obtaining the parametrisation x(θ, φ) = (r(φ) cos θ, r(φ) sin θ, z(φ)).Here we adopt the notation

u1 = θ, u2 = φ.

Theorem 6.6.1. For a surface of revolution we have Γ111 = Γ1

22 = 0,while Γ1

12 =1rdrdφ

, cf. Table 6.6.1.

Proof. For the surface of revolution

x(θ, φ) = (r(φ) cos θ, r(φ) sin θ, z(φ)), (6.6.1)

the metric coefficients are given by the matrix

(gij) =

(r2(φ) 0

0(drdφ

)2+(dzdφ

)2)

5This is atzmit and not pnimit according to Vishne.6Another approach would be to prove the identity 〈xij , xℓ〉 = giℓ,j − gij,ℓ+ gjℓ,i

and use it to prove the theorem.


Γ1ij j = 1 j = 2

i = 1 0 1rdrdφ

i = 2 1rdrdφ

0

Table 6.6.1. Symbols Γ1ij of a surface of revolution (6.6.1)

By Theorem 5.9.4. We will use the formula Γkij =12(giℓ,j−gij,ℓ+gjℓ,i)gℓk

from Theorem 6.5.1. Since the off-diagonal coefficient g12 = 0 vanishes,the diagonal coefficients of the inverse matrix satisfy

gii =1

gii. (6.6.2)

We have∂

∂θ(gii) = 0 since the coefficients gii depend only on the vari-

able φ. Thus the termsgii,1 = 0 (6.6.3)

vanish. Let us now compute the symbols Γ1ij for k = 1. Using formulas

(6.6.2) and (6.6.3), we obtain

Γ111 =

1

2g11(g11,1 − g11,1 + g11,1) by formula (6.6.2)

= 0 by formula (6.6.3).

Similarly, Γ122 =

12g11

(g12,2 − g22,1 + g12,2) =g12,2g11

=ddφ

(0)

g11= 0. Finally,

Γ112 =

1

2g11(g11,2 − g12,1 + g12,1) =

g11,22g11

=

ddφ(r2)

2r2=r dr

dφ

r2,

proving the theorem. �

6.7. Spherical coordinates

Spherical coordinates are useful in understanding surfaces of revo-lution. They were already introduced in Section 5.5.

Definition 6.7.1. Spherical coordinates (ρ, θ, ϕ) in R3 are definedby the following formulas. We have

ρ =√x2 + y2 + z2 =

√r2 + z2

is the distance to the origin, r =√x2 + y2, while ϕ is the angle with

the z-axis, so that cosϕ = zρ. Here θ is the angle inherited from polar

6.8. GREAT CIRCLES 75

coordinates in the x, y plane, so that tan θ = yxwhile x = r cos θ

and y = r sin θ.

Remark 6.7.2. The interval of definition for the variable ϕ is ϕ ∈[0, π] since ϕ = arccos z

ρand the range of the arccos function is [0, π].

Meanwhile θ ∈ [0, 2π] as usual.

Example 6.7.3. The unit sphere S2 ⊆ R3 is defined in sphericalcoordinates by the condition

S2 = {(ρ, θ, ϕ) : ρ = 1}.Definition 6.7.4. A latitude7 on the unit sphere is a circle satis-

fying the equationϕ = constant.

Definition 6.7.5. The equator of the sphere is defined by the equa-tion {ϕ = π

2}.

The equator is the only latitude that is also a great circle (seeSection 6.8) of the sphere.

Each latitude of the sphere is parallel to the equator (i.e., liesin a plane parallel to the plane of the equator). A latitude can beparametrized by setting θ(t) = t and ϕ(t) = constant.

Lemma 6.7.6. On the unit sphere defined by equation {ρ = 1}, ateach point we have the relation

r = sinϕ, (6.7.1)

where r is the distance from the point to the z-axis.

This is immediate from the relation r = ρ sinϕ.

6.8. Great circles

In Section 8.1, we will analyze the geodesic equation. This is asystem of nonlinear second order differential equations. Our goal inthis section is to provide a geometric intuition for this equation. Wewill establish a connection between the following two items:

(1) solutions of this system of ODEs, and(2) great circles on the sphere.

Remark 6.8.1 (Clairaut’s relation as a bridge). Such a connectionis established via the intermediary of Clairaut’s relation for a variablepoint q on a great circle:

r(t) cos γ(t) = const

7kav rochav


where r is the distance from q to the (vertical) axis of revolution and γis the angle at q between the direction of the great circle and thelatitudinal circle; cf. Theorem 7.1.3.

Remark 6.8.2. In Section 7.4, we will verify Clairaut’s relation forgreat circles using synthetic geometry, and also show that the greatcircles satisfy a first order differential equation. In Section 8.1 we willcomplete the connection by deriving Clairaut’s relation from the ge-odesic equation. Thus Clairaut’s relation provides a bridge betweengeometry and analysis.

Let S2 ⊆ R3 be the unit sphere defined by the equation

x2 + y2 + z2 = 1.

Definition 6.8.3. A plane P through the origin is given by anequation

ax+ by + cz = 0, (6.8.1)

where a2 + b2 + c2 > 0.

Definition 6.8.4. A great circle G of S2 is given by an intersection

G = S2 ∩ P.Example 6.8.5 (A parametrisation of the equator of S2). The

equator is parametrized by

α(t) = (cos t)e1 + (sin t)e2.

Theorem 6.8.6. Every great circle can be parametrized by

α(t) = (cos t)v + (sin t)w

where v, w ∈ S2 are orthonormal vectors in R3.

Proof. By the Pythagorean theorem, α(t) is a unit vector andtherefore lies on the unit sphere. �

Example 6.8.7 (an implicit (non-parametric) representation of agreat circle). Recall that we have

x = r cos θ = ρ sinϕ cos θ; y = ρ sinϕ sin θ; z = ρ cosϕ . (6.8.2)

If the circle lies in the plane ax+ by+ cz = 0 where a, b, c are fixed, thegreat circle in coordinates (θ, ϕ) is defined implicitly by the equation

a sinϕ cos θ + b sinϕ sin θ + c cosϕ = 0,

as in (6.8.1).

6.9. POSITION VECTOR ORTHOGONAL TO TANGENT VECTOR OF S2 77

6.9. Position vector orthogonal to tangent vector of S2

Lemma 6.9.1. Let α : R → S2 be a parametrized curve on the

sphere S2 ⊆ R3. Then the tangent vectordα

dtis perpendicular to the

position vector α(t), or in formulas:⟨α(t),

dα

dt

⟩= 0.

Proof. We have 〈α(t), α(t)〉 = 1 by definition of S2. We apply theoperator d

dtto obtain

d

dt〈α(t), α(t)〉 = 0. (6.9.1)

Next, we apply the Leibniz rule (6.4.1) to equation (6.9.1) to obtain⟨α(t),

dα

dt

⟩+⟨dαdt, α(t)

⟩= 2⟨α(t),

dα

dt

⟩= 0,

completing the proof. �

The lemma will be used in the analysis of Clairaut’s relation inSection 7.1.

CHAPTER 7

Clairaut’s relation, geodesic equation

7.1. Clairaut’s relation: bridge bt. geometry and analysis

Our interest in Clairaut’s relation (7.1.1) for curves on the sphere(and more general surfaces of revolution) lies in the motivation it pro-vides for the general geodesic equation on a surface of revolution, as abridge between geometry and analysis.

Remark 7.1.1. We will use Newton’s dot notation for the deriva-tive with respect to t as in α(t), where t is not necessarily an arclengthparameter.

We defined great circles in Section 6.8.

Definition 7.1.2. Given a great circle G ⊆ S2, with paramet-risation α(t), let γ(t) denote the angle, at the point α(t) ∈ S2, betweenthe tangent vector α(t) to the curve and the vector tangent to thelatitude through α(t).

Theorem 7.1.3 (Clairaut’s relation). Let α(t) be a regular para-metrisation of a great circle G on S2 ⊆ R3. Let r(t) denote the distancefrom the point α(t) to the z-axis. Then the following relation holdsalong G:

r(t) cos γ(t) = const. (7.1.1)

Here the constant has value const = rmin, where rmin is the least Eu-clidean distance from a point of G to the z-axis.

The proof exploits the sine law of spherical trigonometry given inSection 7.2.

7.2. Spherical sine law

Let S2 be the unit sphere in 3-space.

Definition 7.2.1. A spherical triangle is the following collectionof data:

(1) the three vertices are points of S2, not lying on a commongreat circle;

(2) each side is an arc of of a great circle;

79

80 7. CLAIRAUT’S RELATION, GEODESIC EQUATION

(3) the angle at each vertex is the angle between tangent vectorsto the sides;

(4) each side has length strictly smaller than π.

Theorem 7.2.2 (Spherical sine law). Let a be the side opposideangle α, let b be the side opposite angle β, and let c be the side oppositeangle γ. Then

sin a

sinα=

sin b

sin β=

sin c

sin γ.

Corollary 7.2.3. If γ = π2then

sin a = sin c sinα. (7.2.1)

Remark 7.2.4. For small values of the sides a, c in a right-angletriangle, we recapture the Euclidean sine law

a = c sinα

as the limiting case of (7.2.1).

Three proofs of the spherical sine law are given in Subsection 7.10.1.

7.3. Longitudes; length-minimizing property

Definition 7.3.1. A longitude1 on S2 is the arc of a great circleconnecting the North Pole e3 = (0, 0, 1)t and the South Pole −e3 =(−1, 0, 0)t.

Next we show that the arc of great circle is length-minimizingamong all paths joining a pair of points on the sphere. Recall thatthe metric coefficients of the sphere in coordinates (θ, φ) are

g11 = sin2 φ,

g22 = 1,

g12 = 0.

Theorem 7.3.2. A shortest spherical path between a pair of pointson S2 is an arc of great circle passing through them.

Proof. Let p0, p1 ∈ S2.

Step 1. Since orthogonal transformations preserve lengths, byapplying a rotation we can assume that p0 and p1 lie on a commonlongitude. To fix ideas, we will assume that this is the longitude inthe xz-plane defined by the coordinate θ = 0 while φ varies from 0to π.

1kav orech

7.3. LONGITUDES; LENGTH-MINIMIZING PROPERTY 81

Step 2. Let x(θ, φ) be the standard parametrisation of the sphereas a surface of revolution. Then pi = x(0, φi), for i = 0, 1.

Step 3. The main point is that moving in the latitudinal directioncan only increase the length of the curve. To implement this technically,consider an arbitrary path α(t), t ∈ [0, 1] joining the two points, sothat α(0) = (0, φ0) and α(1) = (0, φ1). Let β(t) = x(α(t)) be the pathon the sphere. Then pi = β(i) for i = 0, 1. The length L of the path βcan now be bounded from below as follows using Theorem 5.12.1:

L =

∫ 1

0

∣∣∣∣dβ

dt

∣∣∣∣ =∫ 1

0

√sin2 φ

(dθdt

)2+(dφdt

)2

≥∫ 1

0

∣∣∣∣dφ

dt

∣∣∣∣

≥∫ 1

0

dφ

dt

=

∫ φ1

φ0

dφ = φ1 − φ0.

Step 4. The difference φ1−φ0 is precisely the length of the segmentof the longitude between the two points. This proves that the arc of thelongitude containing both points is a shortest path between them. �

Definition 7.3.3. A great circle will be called generic if it doesnot pass through the north and south poles, i.e., does not include alongitude.

Definition 7.3.4. Let pG ∈ S2 be the point of the great circle Gwith the largest z-coordinate among all points of G.

Lemma 7.3.5. Consider a generic great circle G ⊆ S2. Let α(t) bea regular parametrisation of G with α(0) = pG, and denote by α(0) thetangent vector to G at pG. Then G has the following three equivalentproperties:

(1) α(0) is proportional to the vector product e3 × pG;(2) α(0) is perpendicular to the longitude passing through pG;(3) α(0) is tangent at pG to the latitude.

Proof. We will prove item (1).

Step 1. Note that 〈α(0), pG〉 = 0 by Lemma 6.9.1 (tangent vectoris perpendicular to the position vector of the point on the sphere). Itremains to prove that α(0) is orthogonal to e3.


Step 2. Since pG as the point with maximal z-coordinate, thefunction 〈α(t), e3〉 achieves its maximum at t = 0. Hence by Fermat’stheorem we have

d

dt

∣∣∣t=0

〈α(t), e3〉 = 0. (7.3.1)

Step 3. We apply the Leibniz rule to (7.3.1) to obtain⟨dαdt

∣∣∣t=0, e3

⟩= −

⟨α(t),

de3dt

⟩= 0,

since e3 is constant. Thus 〈α(0), e3〉 = 0.2 �

7.4. Preliminaries to proof of Clairaut’s relation

Definition 7.4.1. The spherical distance d(p, q) on S2 is the dis-tance between points p, q ∈ S2 measured along shortest arc of greatcircle passing through them:

d(p, q) = arccos〈p, q〉.By Theorem 7.3.2, d(p, q) is the least length of a path on the sphere

joining p, q.

Lemma 7.4.2. When p be the north pole p = e3, the spherical dis-tance d(e3, q) is the ϕ-coordinate of the point q.

Proof. This follows from the fact that the length of an arc of aunit circle equals the subtended angle. �

As in Section 7.1, we denote by γ(t) the angle formed by the tan-gent vector to a great circle and the latitude. First we note that thecomplementary angle of γ is the angle with the longitude, as follows.

Lemma 7.4.3. Let α(t) be a regular parametrisation of a great circle.Then the following are equivalent:

(1) the angle between α(t) and (the vector tangent to) the latitudeis γ(t);

(2) the angle between α(t) and the vector tangent to the longitudeat the point α(t) is of π

2− γ(t).

2Equation of great circle. This material is optional. One can show thata great circle on the sphere satisfies the equation cot(ϕ) = tan(γ) cos(θ − θ0),where γ is the angle of inclination of the plane (see below). Indeed, a great cir-cle is the intersection of the sphere with a plane through the origin. Let a unitnormal to that plane be u = [− sin(γ), 0, cos(γ)], where for convenience we chooseour x and y axes so that y = 0 (the plane contains the y-axis). Then the equa-tion u · [sinϕ cos θ, sinϕ sin θ, cosϕ] = 0 becomes cot(ϕ) = tan(γ) cos(θ). Rotatingaround the z axis, this becomes cot(ϕ) = tan(γ) cos(θ−θ0) (this gamma has nothingto do with the gamma from Clairaut’s relation).

7.6. DIFFERENTIAL EQUATION OF GREAT CIRCLE 83

Proof. For a surface of revolution x(θ, ϕ) (where u1 = θ and u2 =ϕ), the metric coefficient g12 vanishes (see Section 5.7 and Section 6.6).Hence the tangent vectors x1 =

∂x∂θ

and x2 =∂x∂ϕ

are orthogonal. These

are respectively the tangents to the latitude and the longitude. There-fore the two angles add up to π

2. �

7.5. Proof of Clairaut’s relation

Let us summarize our notation for a great circle G ⊆ S2 withparametrisation β(t).

(1) pG ∈ S2 is the point of G with maximal z-coordinate.(2) ϕ(t) is the spherical coordinate ϕ at the point β(t).(3) r(t) = sinϕ(t) is the distance to the z-axis.(4) ∆t is a spherical triangle with vertices at β(t), pG, and the

north pole e3.(5) The angle of the triangle ∆t at the vertex pG is π

2by Lemma

7.3.5.

We introduce two further arcs:

• ct is the arc of longitude joining the variable point β(t) to e3.• b is the arc of longitude joining pG to e3.

By Lemma 7.4.2, the lengths b and ct are respectively the ϕ-coordinatesof the points pG and β(t). We now apply Corollary 7.2.3 (of the law ofsines) to the right-angle triangle ∆t. We obtain

sin ct sin(π2− γ(t)

)= sin b.

Note that r(t) = sin ct by (6.7.1). Since b is independent of t, we obtainthe relation r(t) cos γ(t) = sin b, proving Clairaut’s relation.

7.6. Differential equation of great circle

Recall that the sphere S2 ⊆ R3 is defined in spherical coordi-nates (θ, ϕ, ρ) by the equation ρ = 1. Note that the parametrisation ofthe great circle G in Clairaut’s relation need not be arclength. Assumethat a great circle G ⊆ S2 is generic (see Definition 7.3.3). Then wecan parametrize G by the value of the spherical coordinate θ. By theimplicit function theorem, we obtain the following lemma.

Lemma 7.6.1. A generic great circle G ⊆ S2 can be defined by asuitable function ϕ = ϕ(θ).


Theorem 7.6.2. Each generic great circle G ⊆ S2 satisfies thefollowing differential equation for ϕ = ϕ(θ):

r2 +

(dϕ

dθ

)2

=r4

const2, (7.6.1)

where r = sinϕ and const = sinϕmin from Theorem 7.1.3.

Proof. An element of length, denoted ds, along the great circle Gdecomposes into a longitudinal (along a longitude, north-south) dis-placement dϕ, and a latitudinal (east-west) displacement rdθ (Leib-niz’s characteristic triangle). Thus ds2 = r2dθ2 + dϕ2 (for details seeProposition 8.2.1). Recall that γ is the angle with the latitude. Hence

{ds sin γ = dϕ

ds cos γ = rdθ

along G. Dividing the first equation by the second, we obtain

tan γ =sin γ

cos γ=

dϕ

r dθ.

Expressing cos2 γ in terms of tan2 γ, we obtain

cos2 γ =1

1 +(dϕr dθ

)2 . (7.6.2)

Substituting into (7.6.2) the value cos γ = constr

from Clairaut’s relation(Theorem 7.1.3), we obtain3

(const

r

)2 (1 +

(dϕrdθ

)2)= 1. (7.6.4)

Multiplying by r4 and dividing by const2 we obtain

r2 +

(dϕ

dθ

)2

=r4

const2where r = sinϕ,


3Equivalently, 1 +(

dϕrdθ

)2=(

r

const

)2or(

dϕrdθ

)2= r2

const2− 1 or dϕ

rdθ=

√r2

const2− 1 or 1

rdϕdθ

=√

r2

const2− 1 or

1

sinϕ

dϕ

dθ=

√sin2 ϕ

const2− 1 (7.6.3)

7.7. METRICS CONFORMAL TO THE FLAT METRIC AND THEIR Γkij 85

This equation is solved explicitly in terms of integrals in Corol-lary 8.2.3 below.4

7.7. Metrics conformal to the flat metric and their Γkij

A particularly important class of metrics are those conformal to theflat metric, in the following sense. We will use the symbol

λ = λ(u1, u2)

for the conformal factor of the metric, as below.

Definition 7.7.1. We say that a metric (gij) is conformal to thestandard flat metric if there is a function λ(u1, u2) > 0 such thatgij(u

1, u2) = λ δij for all i, j = 1, 2.

We will use the notation λi =∂λ∂ui

.

Lemma 7.7.2. Let gij = λ(u1, u2)δij be a metric conformal to thestandard flat metric. Then

Γ111 =

λ12λ,

Γ122 =

−λ12λ,

Γ112 =

λ22λ.

The values of the coefficients are listed in Table 7.7.1.

Proof. The general formula, as in Section 6.5, is

Γkij =1


ℓk.

Step 1. For diagonal metrics the general formula simplifies to

Γkij =1

2(gik,j − gij,k + gjk,i)g

k k. (7.7.1)

4At a point where ϕ is not extremal as a function of θ, the theorem on theuniqueness of solution of ODE applies and gives a unique geodesic through thepoint. However, at a point of maximal ϕ, the hypothesis of the uniqueness theoremdoes not apply. Namely, the square root expression on the right hand side of (7.6.3)does not satisfy the Lipschitz condition as the expression under the square rootsign vanishes. In fact, uniqueness fails at this point, as a latitude (which is not ageodesic) satisfies the differential equation, as well. Here we have r = const, andat an extremal value of ϕ one can no longer solve the equation by separation ofvariables (as this would involve division by the radical expression which vanishes atthe extremal value of ϕ). At this point, there is a degeneracy and general resultsabout uniqueness of solution cannot be applied.


Γ1ij j = 1 j = 2

i = 1 λ12λ

λ22λ

i = 2 λ22λ

−λ12λ

Γ2ij j = 1 j = 2

i = 1 −λ22λ

λ12λ

i = 2 λ12λ

λ22λ

Table 7.7.1. Symbols Γkij of a conformal metric λδij

We have underlined the index k in (7.7.1) to emphasize that no sum-mation is taking place even though k appears as both a subscript anda superscript. Thus

Γkij =1

2gkk(gik,j − gij,k + gjk,i)

where underlining is no longer necessary since the index k appears onlyas a subscript in the formula on the right-hand side.

Step 2. If i = j then the formula simplifies to

Γkii =1

2gkk(gik,i − gii,k + gik,i) =

1

2gkk(2gik,i − gii,k).

By hypothesis, we have g11 = g22 = λ(u1, u2) while g12 = 0 and thelemma follows by examining the cases. If i = j = 1 then

Γ111 =

g11,12λ

=λ12λ.

If i = j = 2 then

Γ122 =

2 · 0− g22,12λ

= −λ12λ.

Similar calculations yield the formulas for the coefficients Γ2ij which are

all listed in the Table. �

Example 7.7.3. An application of these formulas to the hyperbolicmetric appears in Section 12.3.

Definition 7.7.4. Coordinates (u1, u2) with respect to which themetric is expressed by gij = λ(u1, u2)δij are called isothermal coordi-nates.

See Section 10.3 for additional details.

7.8. GRAVITATION, ANTS, AND GEODESICS ON A SURFACE 87

Corollary 7.7.5. A metric in isothermal coordinates (u1, u2) sat-isfies the following identities:{

Γ111 + Γ1

22 = 0

Γ211 + Γ2

22 = 0.

Proof. From Table 7.7.1 we have Γ111 + Γ1

22 =λ12λ

− λ12λ

= 0. Simi-

larly, Γ211 + Γ2

22 = −λ22λ

+ λ22λ

= 0. �

This corollary will be used in Section 10.3 to obtain an equationrelating the Laplacian to the mean curvature.

7.8. Gravitation, ants, and geodesics on a surface

The notion of a geodesic curve is important both in differentialgeometry and in relativity theory. What is a geodesic on a surface? Ageodesic on a surface can be thought of as the path of an ant crawlingalong the surface of an apple, according to the textbook Gravitation[MiTW73, p. 3].

Remark 7.8.1 (Narrow strip of apple’s peel). Imagine that we peeloff a narrow strip5 of the apple’s skin along the ant’s trajectory, andthen lay it out flat on a table. What we obtain is a straight line,revealing the ant’s ability to travel along the shortest path.

On the other hand, a geodesic is defined by a certain nonlinearsecond order ordinary differential equation as in formula (7.9.1). Tomake the geodesic equation more concrete, we will examine the caseof the surfaces of revolution. Here the geodesic equation transformsinto a conservation law (conservation of angular momentum)6 calledClairaut’s relation. The latter lends itself to a synthetic verification forspherical great circles, as we showed in Theorem 7.1.3.

Remark 7.8.2. Thus Clairaut’s relation provides a bridge betweengeometry and ODEs.

In addition to a derivation in Section 7.9, we will also give a longerderivation of the geodesic equation using the calculus of variations.I once heard R. Bott point out a surprising aspect of M. Morse’s foun-dational work in this area. Namely, Morse systematically used thelength functional on the space of curves. The simple idea of using theenergy functional instead of the length functional was not exploiteduntil later. The use of energy simplifies calculations considerably, aswe will see in the optional Section 9.11.2.

5retzua daka6tena’ zaviti


7.9. Geodesic equation on a surface

Consider the following data.

(1) A plane curve Rs→α

R2

(u1,u2)where α = (α1(s), α2(s)).

(2) A parametrisation x : R2 → R3 of a surface M .(3) A curve β = x ◦ α on M given by the composition

Rs→α

R2

(u1,u2)→xR3.

Proposition 7.9.1. Every regular curve β(s) on M satisfies theidentity

β′′ =(αi

′αj

′Γkij + αk

′′)xk +

(Lijα

j ′αi′)n

Proof. Write β = x ◦ α, then β′ = xi(α(s))αi′ by chain rule.

Differentiating again, we obtain

β′′ = dds(xi ◦ α) αi′ + xi α

i′′

= xijαj ′αi

′+ xkα

k ′′.

Meanwhile xij = Γkijxk + Lijn. Thus

β′′ =(Γkijxk + Lijn

)αj

′αi

′+ xkα

k ′′.

Rearranging the terms proves the proposition. �

Definition 7.9.2. A geodesic is a curve β = x◦α on the surfaceMsatisfying one of the following two equivalent conditions is satisfied:

(a) we have for each k = 1, 2,

(αk)′′

+ Γkij(αi)

′

(αj)′

= 0 where′

=d

ds, (7.9.1)

meaning that

(∀k) d2αk

ds2+ Γkij

dαi

ds

dαj

ds= 0; (7.9.2)

(b) the vector β′′ is perpendicular to the surface and one has

β′′ = Lijαi′αj

′n. (7.9.3)

Remark 7.9.3. We will show in Lemma 8.1.1 that such a curve βmust have constant speed.

The equations (7.9.1) will be derived using the calculus of varia-tions, in Section 9.11.2.

7.10. EQUIVALENCE OF DEFINITIONS 89

7.10. Equivalence of definitions

In Section 7.9 we defined a geodesic curve β(s) = x ◦α(s) on a sur-face with parametrisation x = x(u1, u2) in two ways that were claimedto be equivalent:

(a) we have for each k = 1, 2, (αk)′′+ Γkij(α

i)′(αj)

′= 0;

(b) the vector β′′ is normal to the surface and β′′ = Lijαi′αj

′n.

Proof of equivalence. Assume β satisfies (a). We apply Propo-

sition 7.9.1 to the effect that β′′ =(αi

′αj

′Γkij + αk

′′)xk+

(Lijα

j ′αi′)n.

Our assumption implies that the tangent component vanishes and weobtain


′n,

showing that the vector β′′ is normal to the surface.Conversely, suppose β′′ is proportional to the normal vector n. Then

the tangential component of β′′ must vanish, proving the equation(αk)

′′+ Γkij(α

i)′(αj)

′= 0 for each k. �

Corollary 7.10.1. Every geodesic in the plane is a straight line.

Proof. In the plane with a standard parametrisation, all coeffi-cients vanish: Γkij = 0. Then the geodesic equation becomes (αk)

′′= 0.

Integrating twice, we obtain that α(s) is a linear function of s. There-fore the graph is a straight line, proving the corollary. �

7.10.1. Sine law. This material is optional. We present three proofs.

First proof of spherical sine law. Let A,B,C be the angles. Wechoose a coordinate system so that the three vertices of the spherical trian-gle are located at (1, 0, 0), (cos a, sin a, 0) and (cos b, sin b cosC, sin b sinC).The volume of the tetrahedron formed from these three vertices and theorigin is 1

6 sin a sin b sinC. Since this volume is invariant under cyclic rela-beling of the sides and angles, we have sin a sin b sinC = sin b sin c sinA =sin c sin a sinB and therefore sinA

sin a = sinBsin b = sinC

sin c proving the law. Seehttp://math.stackexchange.com/questions/1735860. �

Second proof. Denote byA,B,C the vertices opposite the sides a, b, c.The points A,B,C can be thought of as unit vectors in R3. Note that sin a =|B × C|, etc. Meanwhile sinα = |C ′ × B′| where C ′ is normalisation of theorthogonal component of C when the component parallel to A is eliminated(through the Gram-Schmidt process). Here the component of C parallel to Ais (C · A)A of norm cos b. The orthogonal component is C − (C · A)A is of

norm sin b = |A×C| = |C−(C ·A)A|. Thus we have C ′ = C−(C·A)A|A×C| , B′ =

B−(B·A)A|A×B| . Therefore sinα

sin a =

∣

∣

∣

C−(C·A)A|A×C|

×B−(B·A)A|A×B|

∣

∣

∣

|B×C| = |(C−(C·A)A)×(B−(B·A)A)||A×B| |A×C| |B×C|

http://math.stackexchange.com/questions/1735860


In the denominator we get the product of the three vector products, namely|A×B| |B×C| |C×A|, which is symmetric in the three vectors. Meanwhilein the numerator we get the norm of the vector

(C − (C ·A)A)× (B − (B ·A)A). (7.10.1)

The triple of vectors A,B,C is transformed into the triple A, (C − (C ·A)A), (B − (B · A)A) by a volume-preserving transformation. This com-bined with the fact that A is a unit vector shows that the norm of thevector (7.10.1) equals the absolute value of the determinant of the 3×3 ma-trix [ABC] (i.e., absolute value of the volume of the parallelopiped spannedby the three vectors), which is also symmetric in the three points. Thus we

have sinαsin a = |det[ABC] |

sin a sin b sin c , proving the spherical sine law. �

Third proof of sine law. There is an alternative proof that relieson the identity (a × b) × (a × c) = (a · (b × c))a for each triple of vectors

a, b, c ∈ R3. Indeed, sinα = |(A×B)×(A×C)||A×B| |A×C| = det(ABC)

sin b sin c since A is a unit

vector, and therefore sinαsin a = det(ABC)

sin a sin b sin c as required. �

The sources for the spherical sine law are Smart 1960, pp. 9-10; Gellert et

al. 1989, p. 265; Zwillinger 1995, p. 469. Here (1) Gellert, W.; Gottwald, S.;

Hellwich, M.; Kastner, H.; and Kunstner, H. (Eds.). “Spherical Trigonome-

try.” §12 in VNR Concise Encyclopedia of Mathematics, 2nd ed. New York:

Van Nostrand Reinhold, pp. 261-282, 1989; (2) Smart, W. M. Text-Book on

Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University

Press, 1960; (3) Zwillinger, D. (Ed.). “Spherical Geometry and Trigonom-

etry.” §6.4 in CRC Standard Mathematical Tables and Formulae. Boca

Raton, FL: CRC Press, pp. 468-471, 1995.

CHAPTER 8

Geodesics on surface of revolution, Weingartenmap

8.1. Geodesics on a surface of revolution

Surfaces of revolutionM ⊆ R3 are a rich source of interesting exam-ples of surfaces. Because of the presence of a circle of symmetries, onecan reduce the order of the differential equation of a geodesic from 2to 1, making it easier to solve explicitly. Let x(u1, u2) be a regularparametrisation of M , and let β = x ◦ α be a smooth curve on M .

Lemma 8.1.1. On an arbitrary surface M , a curve β satisfyingthe geodesic equation αk

′′+ Γkijα

i′αj′= 0 as in (7.9.1) is necessarily

constant speed.

Proof. It suffices to prove that the square of the speed has van-ishing derivative. From formula (7.9.3) (namely, β′′ = Lijα

i′αj′n for a

geodesic), we have

d

ds

(|β′|2

)= 2〈β′′, β′〉

= 〈Lijαi′αj ′n , αk ′xk〉= Lijα

i′αj′αk

′〈n, xk〉= 0,


We use the notation u1 = θ, u2 = φ for the parameters in the caseof a surface of revolution generated by a plane curve (r(φ), z(φ)), andwrite x(θ, φ) in place of x(u1, u2). Here the function r(φ) is the distancefrom the point on the surface to the z-axis.

Definition 8.1.2. As in the case of the sphere, the latitude is thecurve on a surface of revolution M obtained by fixing φ = φ0, andparametrized as follows:

β(θ) = x(θ, φ0) = (r(φ0) cos θ, r(φ0) sin θ, z(φ0))

where θ ∈ [0, 2π].

91

92 8. GEODESICS ON SURFACE OF REVOLUTION, WEINGARTEN MAP

Lemma 8.1.3. For a regular unit speed curve β(s) = x(θ(s), φ(s))(not necessarily geodesic) on a surface of revolution M , the angle γ(s)between the curve and the latitude at the point β(s) satisfies the relation

cos γ(s) = rdθ

ds,

where r is the distance from the point β(s) to the z-axis.

Proof. The tangent vector to the latitude is x1 =∂x∂θ(θ, φ). Recall

that g11 = r2 by Theorem 6.6.1, ı.e., |x1| = r; and g12 = 0. Then wecan compute the cosine of the angle between two unit vectors by

cos γ =

⟨x1|x1|

,dβ

ds

⟩.

Recall that by chain rule β′ = x1θ′+ x2φ

′. Thus

cos γ =1

|x1|〈x1, x1θ

′

+ x2φ′

︸︷︷︸chain rule

〉 = θ′

|x1|〈x1, x1〉︸︷︷︸

|x1|2

+φ

′

|x2|〈x1, x2〉︸︷︷︸g12=0

= θ′ |x1| = rθ

′

,


Proposition 8.1.4. On a surface of revolution, the differentialequation of geodesic β(s) = x(θ(s), φ(s)) for k = 1 becomes

r θ′′ + 2dr

dφθ′

φ′

= 0. (8.1.1)

Proof. The Gamma symbols for k = 1 are given by Lemma 6.6.1.

Namely, we have Γ111 = Γ1

22 = 0, while Γ112 =

drdφ

r. We will use the short-

hand notation θ(s), φ(s) respectively for the components α1(s), α2(s)of the curve. Let ′ = d

ds. The differential equation of geodesic β = x◦α

for k = 1 becomes

0 = θ′′ + 2Γ112θ

′φ′ = θ′′ +2 drdφ

rθ′

φ′

,

as required. �

Theorem 8.1.5. On a surface of revolution M ⊆ R3 with coordi-nates (θ, φ), the differential equation of a geodesic for k = 1 is equiva-lent to the first-order differential equation

r2θ′

= const. (8.1.2)

8.2. INTEGRATING GEODESIC EQUATION ON SURFACE OF REV 93

Proof. We multiply formula (8.1.1) by r to obtain

0 = r2θ′′ + 2rdr

dφθ′

φ′

= (r2θ′

)′

by the Leibniz rule and chain rule. Integrating the equation (r2θ′)′= 0

we obtain r2θ′= const as required. �

Corollary 8.1.6. The geodesic equation on a surface of revolutionis equivalent to Clairaut’s relation r(t) cos γ(t) = const where γ is theangle with the latitude; cf. Theorem 7.1.3.

Proof. Since the geodesic β is constant speed by Lemma 8.1.1, wecan assume the parameter s is arclength by rescaling the parameter bya constant factor. Now, by Lemma 8.1.3 we have have

cos γ = rθ′.

Hence by formula (8.1.2) of Theorem 8.1.5, we have

r cos γ = r2θ′ = const.

Thus we obtain r cos γ(s) = const, generalizing Clairaut’s relation fromthe sphere to arbitrary surfaces of revolution. �

Remark 8.1.7. In Section 7.5 we proved Clairaut’s relation syn-thetically (using the spherical law of sines) for the sphere only. Corol-lary 8.1.6 shows that the relation holds for an arbitrary surface of rev-olution in R3.

8.2. Integrating geodesic equation on surface of rev

We have been using the notation (r(φ), z(φ)) for the generatingcurve of a surface of revolution. For the purposes of this section, it willbe helpful to use the notation

{f(φ) = r(φ)

g(φ) = z(φ).

Then we have {g11 = f 2(φ)

g22 =(dfdφ

)2+(dgdφ

)2

In the case of a surface of revolution, the geodesic equation for ageodesic β(s) can be integrated by means of primitives as follows.Let β = x ◦ α(s) be a unit speed geodesic. Thus s is the arclengthparameter and ds is an element of length.


Proposition 8.2.1. Every unit speed curve β(s) on a surface ofrevolution satisfies ds2 = f 2dθ2 + g22dφ

2 or equivalently(ds

dθ

)2

= f 2 + g22

(dφ

dθ

)2

.

Proof. We have

1 =

⟨dβ

ds,dβ

ds

⟩

= 〈x1θ′

+ x2φ′

, x1θ′

+ x2φ′〉

= g11(θ′

)2 + g22(φ′

)2.

Thus,

1 = f 2

(dθ

ds

)2

+ g22

(dφ

ds

)2

. (8.2.1)

Multiplying (8.2.1) by(dsdθ

)2, we obtain

(ds

dθ

)2

= f 2 + g22

(dφ

ds

ds

dθ

)2

= f 2 + g22

(dφ

dθ

)2

,

as required. �

Theorem 8.2.2. The geodesic equation for a surface of revolutioncan be solved explicitly in integrals, producing the following formulafor θ as a function of φ:

θ = c

∫1

f

√√√√(dfdφ

)2+(dgdφ

)2

f 2 − c2dφ+ c′.

Proof. From equation (8.1.2) (equivalent to Clairaut’s relation)we have, replacing r by f :

ds

dθ=f 2

c.

By Proposition 8.2.1, we obtain the formula

f 4

c2= f 2 + g22

(dφ

dθ

)2

(8.2.2)

or, equivalently,

1

c2=

1

f 2+g22f 4

(dφ

dθ

)2

8.3. AREA INTEGRATION IN POLAR COORDINATES 95

(which in the case of the sphere is the equation (7.6.1) of a great circle).Solving equation (8.2.2) for dφ

dθwe obtain

dφ

dθ=

√f4

c2− f 2

g22=f

c

√f 2 − c2

(dfdφ

)2+(dgdφ

)2 .

Solving for θ, we obtain θ = c

∫1f

√( dfdφ)

2+( dg

dφ)2

f2−c2 dφ+ c′. �

Corollary 8.2.3. Along a great circle on S2, we have the followingrelation between the spherical coordinates θ and φ:

θ = c

∫dφ

sinφ√sin2φ − c2

+ c′.

Proof. We have r(φ) = f(φ) = sinφ and z = g(φ) = cosφ. Thus

we obtain θ = c

∫dφ

sinφ√

sin2φ−c2+ c′, which is the equation of a great

circle in spherical coordinates. �

Remark 8.2.4. Integrating this by a substitution u = cotφ weobtain

cotφ = a cos(θ − θ0) (8.2.3)

where a = 1−c2c

.1

8.3. Area integration in polar coordinates

Following some preliminaries on areas, directional derivatives, andHessians, we will deal with a central object in the differential geom-etry of surfaces in Euclidean space, namely the Weingarten map, inSection 8.11.

We review material from calculus on polar, cylindrical, and spher-ical coordinates as regards their role in integration. The polar coor-dinates (r, θ) in the plane arise naturally in complex analysis (of onecomplex variable).

Example 8.3.1 (area element in polar coordinates). Polar coor-dinates (koordinatot koteviot) (r, θ) satisfy r2 = x2 + y2 and x =r cos θ, y = r sin θ. It is shown in elementary calculus that the areaof a region D ⊆ R2 in the plane in polar coordinates is calculated usingthe following area element. The area element of polar coordinates is

dA = r dr dθ,

1The angles θ and φ are switched in the related discussion at http://www.

damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-handout2.pdf

http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-handout2.pdf

http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-handout2.pdf


meaning that the area of D relative to polar coordinates is computedas follows: ∫

D

dA =

∫∫rdrdθ.

8.4. Integration in cylindrical coordinates in R3

Cylindrical coordinates in Euclidean 3-space are studied in VectorCalculus.

Example 8.4.1 (volume element2 in cylindrical coordinates).

(r, θ, z)

are a natural extension of the polar coordinates (r, θ) in the plane.The volume of an open region D ⊆ R3 is calculated with respect tocylindrical coordinates using a suitable volume element. The volumeelement in cylindrical coordinates is

dV = r dr dθ dz,

meaning that the volume of D is calculated as follows:∫

D

dV =

∫∫∫rdr dθ dz.

8.5. Integration in spherical coordinates

Spherical coordinates3 (ρ, θ, ϕ) in Euclidean 3-space are studied inVector Calculus. They were already defined in Section 6.7.

Example 8.5.1 (volume element in spherical coordinates). The co-ordinate ρ is the distance from the point to the origin, satisfying ρ2 =x2 + y2 + z2, or ρ2 = r2 + z2, where r2 = x2 + y2. If we project thepoint orthogonally to the (x, y)-plane, the polar coordinates of its im-age, (r, θ), satisfy x = r cos θ and y = r sin θ. The last coordinate ϕof the spherical coordinates is the angle between the position vectorof the point and the third basis vector e3 in 3-space (pointing upwardalong the z-axis). Thus z = ρ cosϕ while r = ρ sinϕ. Here we havethe bounds 0 ≤ ρ, 0 ≤ θ ≤ 2π, and 0 ≤ ϕ ≤ π (note the different upperbounds for θ and ϕ). the volume of a region in 3-space is calculatedusing the volume element of the form dV = ρ2 sinϕ dρ dθ dϕ, meaningthat the volume of a region D ⊆ R3 is∫

D

dV =

∫∫∫

D

ρ2 sinϕ dρ dθ dϕ .

2koordinatot gliliot3koordinatot kaduriot

8.6. MEASURING AREA ON SURFACES 97

Example 8.5.2 (Area element on the sphere). The area of a spher-ical region D ⊆ S2 on the unit sphere is calculated using an areaelement

sinϕ dθ dϕ .

See also the note.4

8.6. Measuring area on surfaces

Let x(u1, u2) be a parametrisation of a surface M ⊆ R3. The areaof the parallelogram spanned by the tangent vectors x1 and x2 can becalculated as the square root of their Gram matrix, namely the matrixof the first fundamental form. This motivates the following definition.

Definition 8.6.1 (Computation of area). The area of the sur-face M parametrized by x : U → R3 is computed by integrating thearea element √

g11g22 − g212 du1du2 (8.6.1)

over the domain U of the map x. Thus

area(M) =

∫

U

√det(gij) du

1du2,

where M is the region parametrized by x(u1, u2).

Remark 8.6.2. The presence of the square root in the formulais explained in infinitesimal calculus in terms of the Gram matrix,cf. formula (5.6.1). The geometric meaning of the square root is thearea of the parallelogram spanned by the pair of standard (coordinate)tangent vectors.

Example 8.6.3. Consider the parametrisation of the unit sphereprovided by spherical coordinates. Then the integrand is

sinϕ dθ dϕ,

where sinϕ =√det(gij). We recover the formula familiar from calculus

for the area of a region D on the unit sphere:

area(D) =

∫∫

D

sinϕ dθ dϕ,

as in Example 8.5.2.4The area of a spherical region on a sphere of radius ρ = ρ1 is calculated

using the area element dA = ρ21 sinϕ dθ dϕ. Thus the area of a spherical region Don a sphere of radius ρ1 is given by the integral

∫DdA =

∫∫ρ21 sinϕ dθ dϕ =

ρ21∫∫

sinϕ dθ dϕ. An an example, Calculate the area of the spherical region ona sphere of radius ρ1 contained in the first octant (so that all three Cartesiancoordinates are positive).


8.7. Directional derivative as derivative along path

We will represent a vector v ∈ Rn as the velocity vector v = dαdt

of acurve α(t), at t = 0. Typically we will be interested in the case n = 3(or 2).

Definition 8.7.1. Given a function f of n variables, its directionalderivative5 ∇vf at a point p ∈ Rn, in the direction of a vector v isdefined by setting

∇vf =d (f ◦ α(t))

dt

∣∣∣∣t=0

where α(0) = p and α(0) = v.

Lemma 8.7.2. The definition of directional derivative is indepen-dent of the choice of the curve α(t) representing the vector v.

Proof. The lemma is proved in Elementary Calculus [Ke74]. �

Let p = x(u1, u2) be a point of a surface M ⊆ R3. The tangentplane to the surface x = x(u1, u2) at the point p, denoted Tp, wasdefined in Section 5.6 and is the plane passing through p and spannedby vectors x1 and x2:

Tp = SpanR(x1, x2).

It can be thought of as the plane perpendicular to the normal vector nat p.

Definition 8.7.3. Denote by SpanRn the line spanned by n.

Definition 8.7.4 (Orthogonal decomposition). We have an orthog-onal decomposition

R3 = Tp + SpanRn,

meaning that if v ∈ Tp and w ∈ SpanRn then v and w are orthogonal.

Example 8.7.5. Suppose x(u1, u2) is a parametrisation of the unitsphere S2 ⊆ R3. At a point p = (a, b, c) ∈ S2, the normal vector is theposition vector itself:

n =

abc

= p.

In other words, n(u1, u2) = x(u1, u2) = p and we have an orthogonaldecomposition R3 = Tp + SpanR p.

5nigzeret kivunit

8.8. EXTENDING V.F. ALONG SURFACE TO AN OPEN SET IN R3 99

Remark 8.7.6. The sphere is defined implicitly by F (x, y, z) = 0where F (x, y, z) = x2 + y2 + z2 − 1. We have ∇F = (2x, 2y, 2z) andtherefore the gradient∇F at a point is proportional to the radius vectorof the point on the sphere (cf. Lemma 8.8.3).

8.8. Extending v.f. along surface to an open set in R3

Remark 8.8.1. The notion of vector field is defined in infi 4 butonly the maslul of applied mathematics requires it. The maslul of puremathematics takes a course on rings instead.

Now let us return to the set-up

Rt→α

R2

(u1,u2)→xR3.

We consider the curve β = x ◦ α. Let v ∈ Tp be a tangent vector at a

point p ∈ M , defined by v = dβdt

∣∣t=0

, where β(0) = p. By chain rule,we have

v =dαi

dtxi.

Remark 8.8.2. The curve β represents the class of curves withinitial vector v.

Now consider the normal vector to the surface M

n ◦ α(t)along the curve β(t) onM . This vector is only defined along the curve.It is not defined in any open neighborhood in R3. We would like toextend it to a vector field in an open neighborhood of the point p ∈ R3,so as to be able to differentiate it.

Lemma 8.8.3. The gradient ∇F of a function F = F (x, y, z) at apoint where the gradient is nonzero, is perpendicular to the level sur-face {(x, y, z) : F (x, y, z) = 0} of the function F .

This was shown in elementary calculus.Consider a regular parametrisation x(u1, u2) of a neighborhood of

a point p ∈M ⊆ R3 and its normal vector n = n(u1, u2).

Theorem 8.8.4. One can extend n to a differentiable vector fieldN(x, y, z) defined in an open neighborhood of p ∈ R3, in the sense thatwe have

n(u1, u2) = N(x(u1, u2)

). (8.8.1)

Proof. We apply a version of the implicit function theorem forsurfaces to represent the surface implicitly by an equation F (x, y, z) =0, where


(1) the function F is defined in an open neighborhood of p ∈ M ,and

(2) ∇F 6= 0 at p.

By Lemma 8.8.3, the normalisation

N =1

|∇F |∇F

of the gradient ∇F of F gives the required extension in the senseof (8.8.1) of the normal n. �

8.9. Differentiating normal vectors n and N

Our starting point is the following data.

(1) M ⊆ R3 is a surface;(2) x(u1, u2) is a regular parametrisation of M ;(3) n(u1, u2) is the normal vector field along M ;(4) N(x, y, z) is a vector field defined in an open neighborhood

of M in R3 and extends n(u1, u2) so that

n(u1, u2) = N(x(u1, u2)), (8.9.1)

as shown in Section 8.8;(5) ∇v is the directional derivative defined in Section 8.7 via a

representative curve for v;(6) N is a vector-valued function and therefore can be differenti-

ated.

Proposition 8.9.1. Let p ∈M , and v ∈ TpM where v = β′(0) andβ(t) = x(α(t)). Then the directional derivative ∇vN satisfies

∇vN =d (n ◦ α(t))

dt

∣∣∣∣t=0

.

Proof. By (8.9.1), the function n(α(t)) satisfies n(α(t)) = N(β(t)),where β = x ◦ α. By Lemma 8.7.2, the gradient is independent of thechoice of the curve. Hence the directional derivative ∇vN can be cal-culated using the particular curve β as follows:

∇vN =d (N ◦ β(t))

dt=d (n ◦ α(t))

dt,

proving the proposition. �

We will use this proposition to define the Weingarten map in Sec-tion 8.11, after motivating it in terms of the Hessian in Section 8.10.

8.10. HESSIAN OF A FUNCTION AT A CRITICAL POINT 101

8.10. Hessian of a function at a critical point

This section is intended to motivate the definition of the Weingartenmap in Section 8.11. The key observation is Remark 8.10.4 below,relating the Hessian and the Weingarten map.

Proposition 8.10.1. Consider the graph M in R3 of a functionf(x, y) of two variables in the neighborhood of a critical point p of f ,so that ∇f = 0 at p. Then the tangent plane of M at p is a horizontalplane.

Proof. By Theorem 6.1.3, the unit normal vector n is proportionalto (fx, fy,−1)t up to sign. At the critical point, we have fx = fy = 0and therefore n = ±(0, 0, 1)t = e3. �

The Hessian (matrix of second derivatives) of the function at thecritical point captures the main features of the local behavior of thefunction in a neighborhood of the critical point up to negligible higherorder terms. Thus, we have the following typical result concerning thesurface given by the graph of the function in R3.

Theorem 8.10.2. At a critical point p of f , assume that the eigen-values of the Hessian matrix Hf (p) = (fij) are nonzero.

(1) If the eigenvalues of the Hessian have opposite sign at thepoint p, then the graph of f is a saddle point.

(2) If the eigenvalues have the same sign, the graph is a local min-imum or maximum.

Corollary 8.10.3. The determinant of the Hessian determinesthe nature of the critical point p. If detHf < 0 then p is a saddlepoint. If detHf > 0 then p is a local minimum or maximum.

Remark 8.10.4 (Comparison to Weingarten map). Consider theHessian matrix Hf = (fij), i.e., the matrix of second derivatives. Wenow think of the matrix Hf as defining a linear transformation (anendomorphism) of the horizontal plane Tp = R2:

Hf : Tp → Tp.

Here we have w = Hf (v) iff

wi =∑

j

fijvj.

Then the Hessian of a function at a critical point becomes a special caseof theWeingarten map, defined in Section 8.11. The determinant of theWeingarten map plays a special role in determining the geometry of thesurface near p, similar to that of the Hessian noted in Corollary 8.10.3.


8.11. Definition of the Weingarten map

The definition of the Weingarten operator can be viewed as ananalogue of a formula involving the derivative of the normal vectorthat we saw in the context of curves. Note the following points.

(1) (Derivative of normal vector) For a plane curve α(s) with tan-gent vector v(s) and normal vector n(s), we had the equation

d

dsn(s) = ±kα(s)v(s); (8.11.1)

see Proposition 4.3.7. Thus, curvature is closely related to therate of change of the normal vector. The Weingarten map Wp

at a point p ∈M can be thought of as a surface analog of theformula (8.11.1) for curves. NamelyWp carries the informationabout the curvature of M .

(2) In Section 8.10, we considered the special case of a surfacegiven by the graph of a function f of two variables near acritical point of f . Now consider the more general frame-work of a parametrized regular surfaceM in R3 with a regularparametrisation x(u1, u2).

(3) Instead of working with a matrix of second derivatives, we willgive a definition of an endomorphism of the tangent plane.This is basically a coordinate-free way of talking about theHessian matrix.

(4) Theorem 8.8.4 enables us to extend the normal vector field nalong the surface to a vector field N(x, y, z) defined in an openneighborhood of p ∈ R3, so that n(u1, u2) = N(x(u1, u2)).

Definition 8.11.1. Let p ∈ M . Denote by Tp = TpM the tangentplane to the surface at p. The Weingarten map (also known as theshape operator)

Wp : Tp → Tpis the endomorphism of the tangent plane given by directional deriva-tive of the vector-valued function N :

Wp(v) = ∇vN =d

dt

∣∣∣∣t=0

n ◦ α(t), (8.11.2)

where the curve β = x ◦ α is chosen so that β(0) = p while β′(0) = v.

We will show that Wp is a well-defined map in Section 9.1.

CHAPTER 9

Gaussian curvature; second fundamental form

9.1. Properties of Weingarten map

Consider a surface M ⊆ R3. The Weingarten map

Wp : Tp → Tp

at a point p ∈ M was defined in Section 8.11. The following pointsshould be kept in mind:

(1) Suppose p is a critical point of a function f and the surfaceMis the graph of f . Then the Weingarten map of M at p withrespect to the standard coordinates is expressed by the Hessianmatrix of f :

Hf =(fij)i=1,2j=1,2

(see Theorem 9.3.1 for details).(2) The Weingarten map at p measures the way the unit normal

vector to the surface changes as the point p ∈M varies.

Lemma 9.1.1. The map Wp is well defined in that the image isincluded in the tangent plane Tp at p.

Proof. Let v ∈ Tp be a tangent vector. We choose a curve β(t) =

x ◦ α(t) so that v = dβdt

when t = 0. The right hand side of

Wp(v) = ∇vN =d

dt

∣∣∣∣t=0

n ◦ α(t),

as in formula (8.11.2) is a priori a vector in R3. We have to show thatit is indeed produces a vector that lies in the tangent plane Tp i.e., isperpendicular to the unit normal vector n. By Leibniz’s rule,

〈Wp(v), n(α(t))〉 = 〈∇vN, n(α(t))〉

=1

2

d

dt〈n ◦ α(t), n ◦ α(t)〉

=1

2

d

dt(1)

= 0

103

104 9. GAUSSIAN CURVATURE; SECOND FUNDAMENTAL FORM

since n is a unit vector at every point of M . Therefore the imagevector Wp(v) is orthogonal to n. Hence it lies in Tp, as required. �

9.2. Self-adjointness of Weingarten map

Let p ∈M . Recall that the Weingarten mapWp : Tp → Tp is definedas follows. Let v ∈ Tp. Then Wp(v) = ∇vN where N = N(x, y, z) isthe extension to an open neighborhood of p ∈ R3 of the unit normalvector n = n(u1, u2) along M .

The connection with the matrix of second derivatives of the parametri-sation x is given in the following theorem.

Theorem 9.2.1. The map Wp has the following properties:

(1) it is a selfadjoint endomorphism of the tangent plane Tp, and

(2) it satisfies ∀i, j, 〈Wp(xi), xj〉 = −⟨n, ∂2x

∂ui∂uj

⟩.

Proof. By definition of the Weingarten map,∇vN is the derivativeof N along a curve with initial vector v ∈ Tp. To simplify notation,assume that p = x(0, 0).

Step 1. We can choose the particular curve γ(t) = x(t, 0) varyingonly along the first coordinate. Then we have dγ

dt= x1 by definition of

partial derivatives. Therefore

∇x1N =

d

dtN(γ(t)) =

∂n

∂u1

by definition of the directional derivative.

Step 2. Similarly, using the curve δ(t) = x(0, t) varying only alongthe second coordinate, we obtain ∇x2

N = ∂n∂u2

. Thus

∀j = 1, 2, ∇xjN =

∂n

∂uj.

Step 3. We have by Leibniz rule

〈Wp(xi), xj〉 =⟨∂n

∂ui, xj

⟩

=∂

∂ui〈n, xj〉 − 〈n, ∂

∂uixj〉

= −⟨n,

∂2x

∂ui∂uj

⟩.

Thus we obtain that for all i, j,

〈Wp(xi), xj〉 = −⟨n,

∂2x

∂ui∂uj

⟩. (9.2.1)

9.3. RELATION TO THE HESSIAN OF A FUNCTION 105

Step 4. The right-hand side of equation (9.2.1) is an expressionsymmetric in i and j by equality of mixed partials. Thus,

(∀i, j) 〈Wp(xi), xj〉 = 〈xi,Wp(xj)〉.This proves the selfadjointness of Wp by verifying it for a set of basisvectors. �

Corollary 9.2.2. The eigenvalues of the Weingarten map arereal.

Proof. The endomorphism Wp of the vector space TpM is selfad-joint and we apply Corollary 2.3.2. �

Definition 9.2.3. The principal curvatures at p ∈ M , denoted k1and k2, are the eigenvalues of the Weingarten map Wp.

By Corollary 9.2.2, the principal curvatures are real. The principalcurvatures will be discussed in more detail in Section 9.11.

9.3. Relation to the Hessian of a function

The Weingarten map can be understood as a generalisation of theHessian matrix in the following sense.

Theorem 9.3.1 (Relation to the Hessian). Given a function fof two variables, consider its graph M parametrized by x(u1, u2) =(u1, u2, f(u1, u2)). Let p be a critical point of f and n = −e3. Then theinner products 〈Wp(xi), xj〉 at p form the Hessian matrix

Hf (p) =

(∂2f

∂ui∂uj

)

of f at p = x(u1, u2).

Proof. The second partial derivatives of the parametrisation arexij = (0, 0, fij)

t. The normal vector at a critical point is n = −e3.Therefore we obtain 〈xij, n〉 = −fij. The theorem now follows fromformula (9.2.1). �

Example 9.3.2. Consider the plane with its standard parametrisa-tion x(u1, u2) = (u1, u2, 0). We have x1 = e1, x2 = e2, while the normalvector field n

n(u1, u2) =x1 × x2(x1 × x2)

= e1 × e2 = e3

is constant. We can therefore extend it to a constant vector field Ndefined on all of R3. Thus∇vN ≡ 0 andWp(v) ≡ 0, and the Weingartenmap is identically zero. This can be seen also from the vanishing of thesecond derivatives of the parametrisation.


In Sections 9.4 and 9.5 we will present nonzero examples of theWeingarten map.

9.4. Weingarten map of sphere

Theorem 9.4.1. Let M ⊆ R3 be the sphere of radius R > 0. Thenthe Weingarten map at every point p ∈ M of the sphere is the scalarmap Wp : Tp → Tp given by

Wp =1

RIdT

where IdT is the identity map of Tp.

Proof. Represent a tangent vector v ∈ Tp by v = β′(0) whereβ(t) = x ◦ α(t) as usual. On the sphere M of radius R > 0, wehave n(α(t)) = 1

Rβ(t) (normal vector is the normalized position vector).

HenceWp(v) = ∇vN(β(t))

=d

dt

∣∣∣t=0n ◦ α(t)

=1

R

d

dt

∣∣∣t=0β(t)

=1

Rv.

Thus Wp(v) =1

Rv for all v ∈ Tp. In other words, Wp =

1RIdTp . �

Note that the Weingarten map has rank 2 in this case, i.e., it isinvertible.

9.5. Weingarten map of cylinder

For the cylinder, we have the parametrisation

x(u1, u2) = (cosu1, sin u1, u2),

with

x1 = (− sin u1, cos u1, 0)t

x2 = e3,

n = (cos u1, sin u1, 0)t.

Theorem 9.5.1. For the cylinder we have Wp(x1 + cx2) = x1 forall c ∈ R.

9.6. COEFFICIENTS Lij OF WEINGARTEN MAP 107

Proof. Extend n to a vector field N defined in an open neighbor-hood in R3, with the usual relation ∇xiN = ∂

∂uin. Hence we have

∇x1N =∂

∂u1n = (− sin u1, cos u1, 0)t. (9.5.1)

Differentiating with respect to u2, we obtain

∇x2N =∂

∂u2n = (0, 0, 0)t. (9.5.2)

Now let v = vixi be an arbitrary tangent vector. By linearity of direc-tional derivatives, we obtain

∇vN = ∇vixiN = v1∇x1N + v2∇x2N

Hence (9.5.1) and (9.5.2) yield

∇vN = v1∇x1N = v1

− sin u1

cos u1

0

,

and therefore,

Wp(v) = v1

− sin u1

cos u1

0

= v1x1,

where v = vixi. Thus, Wp(x1 + cx2) = x1 for all c ∈ R. �

Note that the Weingarten map has rank 1 in this case.

9.6. Coefficients Lij of Weingarten map

For a regular parametrisation x of a surface M , the vectors (x1, x2)from a basis of the tangent plane Tp. We can therefore exploit theuniqueness of the decomposition with respect to this basis, to definethe coefficients of Wp as follows.

Definition 9.6.1. The coefficients Lij of the Weingarten map aredefined by setting W (xj) = Lijxi = L1

j x1 + L2j x2.

Example 9.6.2. We summarize the the examples developed in theprevious sections.

(1) For the plane, we have (Lij) =

(0 00 0

).

(2) For the sphere, we have (Lij) =

(1r

00 1

r

)= 1

r(δij).

(3) For the cylinder, we have L11 = 1. The remaining coefficients

vanish, so that (Lij) =

(1 00 0

).


9.7. Gaussian curvature

We consider the following data as in the previous sections:

(1) M ⊆ R3 is a surface;(2) x(u1, u2) is a regular parametrisation of a neighborhood in M ;(3) p ∈M is a point of the surface;(4) Tp is the tangent plane to M at p;(5) x1, x2 are tangent vectors forming a basis for Tp;(6) Wp : Tp → Tp is the Weingarten map at p;(7) Li j are the coefficients of Wp defined by Wp(xj) = Li j xi;(8) the antisymmetrisation notation [ ] on indices was defined in

Section 1.6.

Definition 9.7.1. The Gaussian curvature function on M , de-noted K = K(u1, u2), is the determinant of the Weingarten mapat p = x(u1, u2):

K(u1, u2) = det(Wp).

Corollary 9.7.2. We have the formula1

K = det(Lij) = L11L

22 − L1

2L21 = 2L1

[1L22].

Corollary 9.7.3. For both the cylinder and the plane we havethe identity K = 0.2 The Gaussian curvature of a sphere of radius r

is K = det

(1r

00 1

r

)=

1

r2.

Remark 9.7.4 (Sign of Gaussian curvature). Of particular geomet-ric significance is the sign of the Gaussian curvature. The geometricmeaning of negative Gaussian curvature is a saddle point. The geo-metric meaning of positive Gaussian curvature is a point of convexity,such as local minimum or local maximum of the graph of a function oftwo variables.

1 An additional formula for K can be obtained via the signed curvatures ki oforiented curves, defined in Section 11.1 and analyzed in Section 11.6. One can thenassert that Gaussian curvature K of a surface is the product K = k1 · k2 of signedcurvatures k1 and k2 of the pair of curvesM∩P1 andM∩P2, where the planes Pi aredefined by Pi = Span(n, vi) and v1, v2 are orthogonal eigenvectors of the Weingartenmap; see Theorem 11.9.2. We will mostly work with Definition 9.7.1 of Gaussiancurvature; we mention the additional definition in terms of signed curvatures ofcurves for general culture as it was the original definition by Gauss.

2cf. Table 11.11.1.

9.8. SECOND FUNDAMENTAL FORM 109

9.8. Second fundamental form

Recall that we extended the normal vector field n(u1, u2) along thesurfaceM ⊆ R3 near a point p ∈M , to a smooth vector field N(x, y, z)defined in an open neighborhood of p viewed as a point of R3 sothat N(x(u1, u2)) = n(u1, u2).

Definition 9.8.1. The second fundamental form IIp is the bilinearform on the tangent plane Tp defined for u, v ∈ Tp by

IIp(u, v) = −〈∇uN, v〉. (9.8.1)

Remark 9.8.2 (Surface geodesics as space curves). What does thesecond fundamental form measure? In the first approximation, one cansay that the second fundamental form measures the curvature of spacecurves defined by

geodesics on M viewed as curves of R3 (cf. 4.2.3)

as illustrated by Theorem 9.9.1 below.

Remark 9.8.3. The minus sign in formula (9.8.1) is just a conven-tion, to make some later formulas, such as (9.8.2), to work out better.It is important to note the following:

this sign does not affect the sign of the Gaussian curvature!

Definition 9.8.4. The coefficients Lij of the second fundamentalform are defined to be

Lij = IIp(xi, xj) = −⟨∂n

∂ui, xj

⟩.

Lemma 9.8.5. The coefficients Lij of the second fundamental formare symmetric in i and j, more precisely

Lij = +〈xij, n〉. (9.8.2)

Thus we have L12 = L21.

Proof. We have 〈n, xi〉 = 0 everywhere. Hence ∂∂uj

〈n, xi〉 = 0, i.e.⟨

∂

∂ujn, xi

⟩+ 〈n, xij〉 = 0

or

〈n, xij〉 = −〈∇xjN, xi〉 = +IIp(xj, xi)

and the proof is concluded by the equality of mixed partials (Clairaut–Schwarz theorem). �


Corollary 9.8.6. The second fundamental form enables us toidentify the normal component of the second partial derivatives of theparametrisation x(u1, u2). In formulas,

xij = Γkijxk + Lijn.

Proof. We write xij = Γkijxk + cn. Now form the inner productwith n to obtain Lij = 〈xij, n〉 = 0 + c〈n, n〉 = c. �

9.9. Geodesics and second fundamental form

The second fundamental form measures the curvature of geodesicson M viewed as space curves, in the following sense.

Theorem 9.9.1. Let β(s) be a unit speed geodesic on a surfaceM ⊆R3, so that β′(s) ∈ TpM at a point p = β(s). Then the absolute value ofthe second fundamental form applied to the pair (β′, β′) is the curvatureof β at p:

|IIp(β′, β′)| = kβ(s),

where kβ is the curvature of the curve β viewed as a curve in R3.

Proof. We apply formula (7.9.3) for the curvature of a curve.Since |n| = 1, we have

kβdef= |β′′| =

∣∣∣Lijαi′αj ′∣∣∣ . (9.9.1)

On the other hand, recall that β = x ◦ α. We have

IIp(β′, β′) = IIp(xiα

i′, xjαj ′) = αi

′αj

′IIp(xi, xj) = αi

′αj

′Lij.

This expression coincides with (9.9.1) up to sign, completing the proof.�

Proposition 9.9.2 (Lowering an index). We have the followingrelation between the coefficients of the second fundamental form andthe Weingarten map:

Lij = −Lkjgki.Proof. By definition,

Lij = 〈xij, n〉 = −⟨

∂

∂ujn, xi

⟩= −〈Lkjxk, xi〉 = −Lkjgki


Corollary 9.9.3 (Raising an index). We have the following rela-tion between the coefficients of the Weingarten map and the coefficientsof the second fundamental form: Li j = −gikLkj.

9.10. THREE FORMULAS FOR GAUSSIAN CURVATURE 111

Proof. We start with the relation Lij = −Lkjgki of Proposi-tion 9.9.2. We now multiply the relation on both sides by giℓ, andsum over i, obtaining

Lijgiℓ = −Lkjgkigiℓ

= −Lkjδℓk= −Lℓj

as required. �

9.10. Three formulas for Gaussian curvature

Theorem 9.10.1. We have the following three equivalent formulasfor the Gaussian curvature:

(a) K = det(Lij) = 2L1[1L

22];

(b) K = det(Lij)/det(gij);(c) K = − 2

g11L1[1L

22].

Proof. Here formula (a) is our definition of K. To prove for-mula (b), we use the formula

Lij = −Lkjgki (9.10.1)

of Proposition 9.9.2. By the multiplicativity of determinant with re-spect to matrix multiplication,

det(Lij) = (−1)2det(Lij)

det(gij).

Let us now prove formula (c). The proof is a calculation. Note thatby definition of the antisymmetrisation notation, 2L1[1L

22] = L11L

22 −

L12L21. Applying (9.10.1), we obtain

− 2

g11

(L1[1L

22]

)=

1

g11

((L1

1g11 + L21g21

)L2

2 −(L1

2g11 + L22g21

)L2

1

)

=1

g11

(g11L

11L

22 + g21L

21L

22 − g11L

12L

21 − g21L

22L

21

)

= det(Li j) = K

as required. �

Remark 9.10.2. In the proof above, we could calculate using eitherthe formula Lij = −Lkjgki, or the formula Lij = −Lkigkj. Only theformer one leads to the appropriate cancellations as above.


9.11. Principal curvatures of surfaces

We start with the following data.

(1) a point p ∈M of a surface M .(2) Wp : Tp → Tp is an endomorphism of the tangent plane TpM .(3) The principal curvatures at p were defined in Sections 8.11

and 9.2.

Let us recall the definition of the principal curvatures.

Definition 9.11.1. The principal curvatures of M at the point p,denoted k1 and k2, are the eigenvalues of the Weingarten map Wp.

Example 9.11.2. Let M be the hyperbolic paraboloid given bythe graph in R3 of the function f(x, y) = ax2 − by2, a, b > 0 (seeSection 3.6). The origin p = (0, 0) is a critical point of f . Thereforethe matrix of the Weingarten map Wp at p is the Hessian matrix Hf

of f , namely Hf =

(2a 00 −2b

). Therefore the principal curvatures at

the origin are k1 = 2a and k2 = −2b (the order of the two eigenvaluesis immaterial).

Remark 9.11.3. The curvatures k1 and k2 are necessarily real, bythe selfadjointness ofW (Theorem 9.2.1) together with Corollary 2.3.2.

Theorem 9.11.4. The Gaussian curvature K(u1, u2) of M at apoint p = x(u1, u2) equals the product of the principal curvatures at p.

Proof. The determinant of a 2 by 2 matrix equals the productof its eigenvalues: K = det(Lij) = k1k2, where (Lij) is the matrixrepresenting the endomorphism Wp. �

Recall that IIp denotes the second fundamental form at p ∈M . Weproved in Theorem 9.9.1 that

|IIp(β′, β′)| = kβ, (9.11.1)

where kβ is the curvature of the unit speed geodesic β(s) with velocityvector β′. We now apply this result to the eigendirections.

Theorem 9.11.5. Let v ∈ TpM be a unit eigenvector belonging to aprincipal curvature k at the point p ∈M . Let β(s) be a geodesic on Msatisfying β(0) = p, β′(0) = v. Then the curvature of β as a spacecurve is the absolute value of k :

kβ(0) = |k|.

9.11. PRINCIPAL CURVATURES OF SURFACES 113

Proof of Theorem 9.11.5. Since v is an eigenvector of Wp, weobtain from (9.11.1) that

kβ(0) = |IIp(β′, β′)|= |⟨Wp (β

′), β′⟩ |= |〈kβ′, β′〉|= |k〈β′, β′〉|= |k|


Corollary 9.11.6. The absolute value of the Gaussian curvatureat a point p of a regular surface in R3 is the product of curvatures oftwo perpendicular geodesics passing through p, whose tangent vectorsare eigenvectors of the Weingarten map at the point.

Remark 9.11.7. The absolute value can be removed. Choose a unitnormal n to the surface M at p ∈ M . Consider geodesics βi(s) suchthat β′

1(0) and β′2(0) are the principal directions at p. We can define

the signed curvature ki of geodesics βi(s) by setting ki = 〈βi, n〉. ThenK = k1k2 (without absolute value bars). See further in Theorem 11.9.2.

9.11.1. Normal and geodesic curvatures. The material in this sec-tion is optional. Let x(u1, u2) be a surface in R3. Let α(s) = (α1(s), α2(s))a curve in R2, and consider the curve β = x ◦ α on the surface x. Con-sider also the normal vector to the surface, denoted n. The tangent unitvector β′ = dβ

ds is perpendicular to n, so β′, n, β′ × n are three unit vector

spanning R3. As β′ is a unit vector, it is perpendicular to β′′, and there-fore β′′ is a linear combination of n and β′×n. Thus β′′ = knn+kg(β

′×n),where kn, kg ∈ R are called the normal and the geodesic curvature (resp.).Note that kn = β′′ · n, kg = β′′ · (β′ × n). If k is the curvature of β, then wehave that k2 = ||β′′||2 = k2n + k2g .

Exercise 9.11.8. Let β(s) be a unit speed curve on a sphere of radius r.Then the normal curvature of β is ±1/r.

Remark 9.11.9. Let π be a plane passing through the center of the sphereS in Exercise 9.11.8, and Let C = π ∩S. Then the curvature k of C is 1/r,and thus kg = 0. On the other hand, if π does not pass through the centerof S (and π ∩ S 6= ∅) then the geodesic curvature of the intersection 6= 0.

Proposition 9.11.10. If a unit speed curve β on a surface is geodesicthen its geodesic curvature is kg = 0.

Proof. If β is a geodesic, then β′′ is parallel to the normal vector n, soit is perpendicular to n× β′′, and therefore kg = β′′ · (n× β′′) = 0. �


Exercise 9.11.11. The inverse direction of Proposition 9.11.10 is alsocorrect. Prove it.

Example 9.11.12. Any line γ(t) = at+ b on a surface is a geodesic, asγ′′ = 0 and therefore kg = 0.

Example 9.11.13. Take a cylinder x of radius 1 and intersect it with aplane π parallel to the xy plane. Let C = x ∩ π - it is a circle of radius 1.Thus k = 1. Show that kn = 1 and thus C is a geodesic.

The principal curvatures, denoted k1 and k2, are the eigenvalues of theWeingarten map W . Assume k1 > k2.

Theorem 9.11.14. The minimal and maximal values of the absolutevalue of the normal curvature |kn| at a point p of all curves on a surfacepassing through p are |k2| and |k1|.

Proof. This is proven using the fact that kg = 0 for geodesics. �

9.11.2. Calculus of variations and the geodesic equation. Thematerial in this subsection is optional. Calculus of variations is known astachsiv variatsiot. Let α(s) = (α1(s), α2(s)), and consider the curve β =x ◦ α on the surface given by [a, b]s→αR2→xR3. Consider the energy

functional E(β) =∫ ba ‖β′‖2ds. Here ′ = d

ds . We have dβds = xi(α

i)′

by chainrule. Thus

E(β) =∫ b

a〈β′, β′〉ds =

∫ b

a

⟨xiα

i′, xjαj ′⟩ds =

∫ b

agijα

i′αj′ds. (9.11.2)

Consider a variation α(s) → α(s)+ tδ(s), t small, such that δ(a) = δ(b) = 0.If β = x ◦α is a critical point of E , then for any perturbation δ vanishing atthe endpoints, the following derivative vanishes:

0 =d

dt

∣∣∣∣∣t = 0

E(x ◦ (α+ tδ))

=d

dt

∣∣∣∣∣t = 0

{∫ b

agij(α

i + tδi)′

(αj + tδj)′

ds

}(from equation (9.11.2))

=

∫ b

a

∂ (gij ◦ (α+ tδ))

∂tαi

′

αj′

ds

︸︷︷︸A

+

∫ b

agij(αi

′

δj′

+ αj′

δi′)ds

︸︷︷︸B

,

so that we haveA+B = 0. (9.11.3)

We will need to compute both the t-derivative and the s-derivative of thefirst fundamental form. The formula is given in the lemma below.

Lemma 9.11.15. The partial derivatives of gij = gij ◦ (α(s) + tδ(s))

along β = x ◦ α are given by the following formulas: ∂∂t(gij ◦ (α + tδ)) =

(〈xik, xj〉+ 〈xi, xjk〉)δk, and ∂gik∂s = (〈xim, xk〉+ 〈xi, xkm〉)(αm)

′.

9.11. PRINCIPAL CURVATURES OF SURFACES 115

Proof. We have∂

∂t(gij ◦ (α+ tδ)) =

∂

∂t〈xi ◦ (α+ tδ)), xj ◦ (α+ tδ)〉

=

⟨∂

∂t(xi ◦ (α+ tδ)), xj

⟩+

⟨xi,

∂

∂t(xj ◦ (α+ tδ))

⟩

= 〈xikδk, xj〉+ 〈xi, xjkδk〉 = (〈xik, xj〉+ 〈xi, xjk〉)δk.Furthermore, g′ik = 〈xi, xk〉′ = 〈xim(αm)′ , xk〉+ 〈xi, xkm(αm)

′〉. �

Lemma 9.11.16. Let f ∈ C0[a, b]. Suppose that for all g ∈ C0[a, b] we

have∫ ba f(x)g(x)dx = 0. Then f(x) ≡ 0. This conclusion remains true if

we use only test functions g(x) such that g(a) = g(b) = 0.

Proof. We try the test function g(x) = f(x). Then∫ ba (f(x))

2ds = 0.

Since (f(x))2 ≥ 0 and f is continuous, it follows that f(x) ≡ 0. If wewant g(x) to be 0 at the endpoints, it suffices to choose g(x) = (x− a)(b−x)f(x). �

Theorem 9.11.17. Suppose β = x ◦ α is a critical point of the en-ergy functional (endpoints fixed). Then β satisfies the differential equation

(∀k) (αk)′′+ Γkij(α

i)′(αj)

′= 0.

We use Lemma 9.11.15 to evaluate the term A from equation (9.11.3) as

follows:A =∫ ba (〈xik, xj〉+〈xi, xjk〉)(αi)

′

(αj)′

δkds = 2

∫ b

a〈xik, xj〉αi

′

αj′

δkds,

since summation is over both i and j. Similarly, B = 2∫ ba gijα

i′

δj′

ds =

−2∫ ba

(gijα

i′)′

δjds by integration by parts, where the boundary term van-

ishes since δ(a) = δ(b) = 0. Hence B = −2∫ ba

(gikα

i′)′δkds by changing an

index of summation. Thus 12ddt

∣∣∣∣∣t=0

(E) =∫ ba

{〈xik, xj〉αi′αj ′ −

(gikα

i′)′}

δkds.

Since this is true for any variation (δk), by Lemma 9.11.16 we obtain the

Euler-Lagrange equation (∀k) 〈xik, xj〉αi′αj ′ −(gikα

i′)′

≡ 0, or

〈xik, xj〉αi′αj ′ − gik′αi

′ − gikαi′′ = 0. (9.11.4)

Using the formula from Lemma 9.11.15 for the s-derivative of the first fun-damental form, we can rewrite the formula (9.11.4) as follows:

0 = 〈xik, xj〉αi′

αj′

− 〈xim, xk〉αm′αi

′

− 〈xi, xkm〉αi′

αm′ − gikα

i′′

= −〈Γnimxn, xk〉αm′αi

′

− gikαi′′

= −Γnimgnkαm′αi

′

− gikαi′′

,

where the cancellation of the first and the third term in the first line results

from replacing index i by j and m by i in the third term. This is true ∀k.


Now multiply by gjk to obtain gjkΓnimgnkαm′αi

′

+gjkgikαi′

= δjnΓnimαm′αi

′

+

δjiαi′′

= Γjimαm′αi

′

+ αj′′= 0, which is the desired geodesic equation.

CHAPTER 10

Minimal surfaces; Theorema egregium

10.1. Mean curvature, minimal surfaces


(1) p ∈M a point on a surface M ;(2) Tp, the tangent plane at p;(3) Wp : Tp → Tp the Weingarten map, an endomorphism of Tp.

Definition 10.1.1. The mean curvature H = H(u1, u2) of a sur-faceM ⊆ R3 at a point p = x(u1, u2) is half the trace of the Weingartenmap: H = 1

2trace(Wp) =

12(k1 + k2) =

12Lii.

Definition 10.1.2. A surface M ⊆ R3 is called minimal if H = 0at every point, i.e. k1 + k2 = 0.

Remark 10.1.3 (Meaning of sign). The sign of the mean curvaturehas no geometric meaning and depends on the choice of normal vector n(from among the pair n,−n) used in the definition of Wp and IIp. Thisis in contrast with Gaussian curvature of M where the sign does havegeometric meaning.

See Table 11.11.1 on plane and cylinder for comparison of the geo-metric meaning of mean curvature and Gaussian curvature.

Remark 10.1.4 (Area minimisation). Geometrically, a minimalsurface is represented locally by a soap film.1 A soap film seeks tominimize area locally. In this sense minimal surfaces are similar togeodesics, which minimize length locally.

10.2. Scherk surface

The Scherk surface is a well-known minimal surface which is peri-odic in the sense explained below. The following lemma will be usefulin the study of the Scherk surface.

Definition 10.2.1. The Scherk surface is the graph of z = ln cos ycosx

.

1krum sabon, as opposed to bu’at sabon. Dip (tvol) a wire (tayil) into soapywater (mei sabon).

117

118 10. MINIMAL SURFACES; THEOREMA EGREGIUM

Figure 10.2.1. Scherk surface

Thus the Schirk surface is parametrized by(x, y, ln

(cos ycosx

)). Since

cosine is periodic, so is the parametrisation in both variables.

Theorem 10.2.2 (Scherk surface). The Scherk surface is a minimalsurface.

Proof. Step 1. The Scherk surface by definition is parametrizedby the map x(x, y) = (x, y, ln cos y− ln cos x). Clearly, we have x12 = 0.Therefore L12 = 〈x12, n〉 = 0. Thus the matrix (Lij) is diagonal.

Step 2. By Corollary 9.9.3, the mean curvature H satisfies the formula2H = traceWp = −(L11g

11+L22g22) and therefore the condition H = 0

is equivalent toL11g

11 + L22g22 = 0. (10.2.1)

Let g = det(gij) (note that g12 6= 0). By definition of the inversematrix, g11 = g22

gand g22 = g11

g. Thus the condition (10.2.1) becomes

L11g22 + L22g11g

= 0

where g 6= 0. Thus we have the following reformulation of minimalityfor the Scherk surface M :

M is minimal ↔ L11

g11+L22

g22= 0. (10.2.2)

Step 3. We have {x1 = (1, 0,− tan x)t,

x2 = (0, 1, tan y)t.(10.2.3)

10.3. MINIMAL SURFACES IN ISOTHERMAL COORDINATES 119

Hence

g11 = 1 + tan2 x = sec2 x, g22 = 1 + tan2 y = sec2 y. (10.2.4)

The normal vector is the normalisation of the cross product x1 × x2,

i.e., of (− tan x, tan y, 1)t. Let C =√1 + tan2 x+ tan2 y, so that

n = 1C(tan x,− tan y, 1)t.

Step 4. We have x11 = (0, 0,− sec2(x))t and x22 = (0, 0, sec2 y)t.Therefore {

L11 = 〈n, x11〉 = − sec2 xC,

L22 = 〈n, x22〉 = sec2 yC.

(10.2.5)

Thus by (10.2.4), L11

g11= − 1

Cand L22

g22= 1

C, and therefore L11

g11+ L22

g22= 0.

Hence by (10.2.2), M is minimal, proving the minimality of the Scherksurface. �

See Figure 10.2.1 for Scherk surface.

Remark 10.2.3. The standard parametrisation(x, y, ln cos y

cosx

)of the

Scherk surface provides an example of a parametrisation where thesecond fundamental form is given by a diagonal matrix, but the firstfundamental form and the Weingarten map are given by nondiagonalmatrices.

See Figure 10.2.2 for another example of a minimal surface, calledthe helicoid.

10.3. Minimal surfaces in isothermal coordinates

In this section we study the PDE of a minimal surface M ⊆ R3.The following definition recalls Definition 7.7.4.

Definition 10.3.1. A parametrisation x(u1, u2) is called isothermalif the following two equivalent conditions are satisfied:

(1) there is a function f = f(u1, u2) > 0 such that gij = f 2δij,(2) 〈x1, x1〉 = 〈x2, x2〉 and 〈x1, x2〉 = 0.

Definition 10.3.2. The function f 2 is called the conformal factorof the metric.

Sometimes the function f itself is called the conformal factor. Theusage will be clear from context.


Figure 10.2.2. Helicoid: a minimal surface

Proposition 10.3.3. Consider a surfaceM ⊆ R3. Assume x(u1, u2)is a parametrisation of M in isothermal coordinates. Then it satisfiesthe partial differential equation

∂2x

∂(u1)2+

∂2x

∂(u2)2= −2f 2Hn,

where H = H(u1, u2) is the mean curvature and n = n(u1, u2) is thenormal vector to M ⊆ R3.

Proof. Step 1. We use the formula xij = Γkijxk + Lijn to write

∆ x = x11 + x22

= Γ111x1 + Γ2

11x2 + L11n+ Γ122x1 + Γ2

22x2 + L22n

=(Γ111 + Γ1

22

)x1 +

(Γ211 + Γ2

22

)x2 + (L11 + L22)n.

By Corollary 7.7.5, with respect to isothermal coordinates we necessar-ily have the identities Γ1

11 + Γ122 = 0 and Γ2

11 + Γ222 = 0. Thus we have

∆ x = (L11 + L22)n.

10.4. MINIMALITY AND HARMONIC FUNCTIONS 121

Step 2. Recall that with respect to isothermal coordinates, wehave Lii = −f 2Lii. Therefore ∆ x = (L11 +L22)n = −(L1

1 +L22)f

2n =

−2Hf 2n as required.2 �

10.4. Minimality and harmonic functions

In this section we study the relation between minimal surfaces andharmonic functions. The latter are familiar from complex functiontheory. We first recall a definition from Section 4.6, already used inSection 10.3.

Definition 10.4.1. Assume F = F (u1, u2) is twice differentiable.

Then the Laplacian of F , denoted ∆(F ), is ∆F = ∂2F∂(u1)2

+ ∂2F∂(u2)2

.

Definition 10.4.2. We say that F is harmonic if ∆F = 0.

Harmonic functions are important in the study of heat flow, or heattransfer.3

Theorem 10.4.3. Let M ⊆ R3 be a surface with parametrisa-tion x(u1, u2) = (x(u1, u2), y(u1, u2), z(u1, u2)) Assume that the coor-dinates (u1, u2) are isothermal. Then M is minimal if and only if thecoordinate functions x, y, z are harmonic.

Proof. From Proposition 10.3.3 we have

|∆(x)| =√(∆x)2 + (∆y)2 + (∆z)2 = 2f 2|H|.

Therefore the Laplacian of x vanishes if and only if ∆x = ∆y = ∆z = 0,which occurs if and only if H = 0. �

2An alternative proof following Do Carmo [Ca76]. This material is optional.

Note that Lij = −Lmjgmi = −Lm

jf2δmi = −f2Li

j . Thus Lij = −Lij

f2so that the

mean curvature H satisfies

H =1

2Li

i = −L11 + L22

2f2. (10.3.1)

Differentiating 〈x1, x2〉 = 0 we obtain ∂∂u2 〈x1, x2〉 = 0. Therefore

〈x12, x2〉+ 〈x1, x22〉 = 0. (10.3.2)

So −〈x12, x2〉 = 〈x1, x22〉. By Definition 10.3.1 of isothermal coordinates, we have

〈x1, x1〉 − 〈x2, x2〉 = 0. (10.3.3)

Differentiating (10.3.3) and applying (10.3.2), we obtain 0 = ∂∂u1 〈x1, x1〉 −

∂∂u1 〈x2, x2〉 = 2〈x11, x1〉 − 2〈x21, x2〉 = 2〈x11, x1〉 + 2〈x22, x1〉 = 2〈x11 + x22, x1〉.Inspecting the u2-derivatives, we similarly obtain 〈x11 + x22, x2〉 = 0. Since x1, x2and n form an orthogonal basis, the sum x11+x22 is proportional to n. Write x11+x22 = cn. Applying (10.3.1), we obtain c = 〈x11 + x22, n〉 = 〈x11, n〉 + 〈x22, n〉 =L11 + L22 = −2f2H, as required.

3ma’avar chom.


Theorem 10.4.4. Let a > 0. The catenoid parametrized by

x(θ, φ) = (a coshφ cos θ, a coshφ sin θ, aφ),

is a minimal surface.

Proof. The generating curve is the curve r(φ) = a coshφ andz(φ) = aφ.4 Then according to the general formula, g11 = r2 =a2 cosh2 φ. Also,

g22 = ( drdφ)2 + ( dz

dφ)2 = (a sinhφ)2 + a2 = a2 cosh2 φ = g11,

and g12 = 0. We conclude that the coordinates (θ, φ) are isothermal.Finally,

x11 + x22 = (−a coshφ cos θ,−a coshφ sin θ, 0)t

+ (a coshφ cos θ, a coshφ sin θ, 0)t

= (0, 0, 0).

By Theorem 10.4.3, we have H = 0 and therefore the catenoid is aminimal surface. �

See Figure 10.4.1.

Figure 10.4.1. Catenoid: a minimal surface

4This is the catenary curve expressing the shape of suspension bridges.

10.5. INTRO TO THEOREMA EGREGIUM; RIEMANN’S FORMULA 123

Remark 10.4.5. The catenoid is the only complete surface of rev-olution which is minimal [We55, p. 179], other than the plane. Thehelicoid and the catenoid are locally isometric.

10.5. Intro to theorema egregium; Riemann’s formula

Understanding the intrinsic nature of Gaussian curvature, i.e. thetheorema egregium of Gauss, clarifies the geometric classification ofsurfaces. A historical account and an analysis of Gauss’s proof of thetheorema egregium may be found in [Do79].5

We will present a compact formula for Gaussian curvature in Sec-tion 10.7, and its proof, along the lines of the argument in Manfredodo Carmo’s book [Ca76].6 The appeal of an old-fashioned, computa-tional, coordinate notation proof is that it obviates the need for higherorder objects such as connections, tensors, exponental map, etc, andis, hopefully, directly accessible to a student not yet familiar with thesubject (see note 12.11.1).

Remark 10.5.1. We would like to distinguish two types of proper-ties of a surface in Euclidean space: intrinsic and extrinsic.

Understanding the extrinsic/intrinsic dichotomy is equivalent to un-derstanding the theorema egregium of Gauss. The theorema egregium isthe key insight lying at the foundation of differential geometry as con-ceived by Bernhard Riemann in his essay [Ri1854] presented beforethe Royal Scientific Society of Gottingen in 1854 (see Section 10.5).

The theorema egregium asserts that an infinitesimal invariant of asurface in Euclidean space, called Gaussian curvature, is an “intrinsic”invariant of the surface M . In other words, Gaussian curvature of Mis independent of its isometric embedding in Euclidean space. Thistheorem paves the way for an intrinsic definition of curvature in modernRiemannian geometry.

Remark 10.5.2. We will prove that Gaussian curvature K is anintrinsic invariant of a surface in Euclidean space, in the followingprecise sense: K can be expressed in terms of the coefficients of thefirst fundamental form and their derivatives alone.

5Our formula (10.7.2) for Gaussian curvature is similar to the traditional for-mula for the Riemann curvature tensor in terms of the Levi-Civita connection (notethe antisymmetrisation in both formulas, and the corresponding two summands),without the burden of the connection formalism.

6Other undergraduate differential geometry textbooks arediscussed at https://mathoverflow.net/questions/7834/

undergraduate-differential-geometry-texts

https://mathoverflow.net/questions/7834/undergraduate-differential-geometry-texts

https://mathoverflow.net/questions/7834/undergraduate-differential-geometry-texts


A priori the possibility of expressing K in such a fashion is notobvious, as the naive definition of K is in terms of the Weingartenmap.

The intrinsic nature of Gaussian curvature paves the way for a tran-sition from classical differential geometry, to a more abstract approachof modern differential geometry (see also Subsection 13.4). The dis-tinction can be described roughly as follows. Classically, one studiessurfaces in Euclidean space. Here the first fundamental form (gij) ofthe surface is the restriction of the Euclidean inner product. Mean-while, abstractly, a surface comes equipped with a set of coefficients,which we deliberately denote by the same letters, (gij) in each coordi-nate patch, or equivalently, its element of length. One then proceedsto study its geometry without any reference to a Euclidean embedding,cf. (16.7.2).

Such an approach was pioneered in higher dimensions in Riemann’sessay. The essay contains a single formula [Ri1854, p. 292]. See Spivak[Sp79, p. 149]), where the formula appears on page 159. This is theformula for the element of length of a surface of constant (Gaussian)curvature K ≡ α:

1

1 + α4

∑x2

√∑dx2 (10.5.1)

where today, of course, we would incorporate a summation index i aspart of the notation, as in

ds =1

1 + α4

∑i(x

i)2

√∑

i

(dxi)2 (10.5.2)

cf. formulas (16.7.3), (16.20.1).We will explain the notation dxi in Section 13.9.

Remark 10.5.3. As noted in earlier sections, given a surface M in3-space which is the graph of a function of two variables, at a criticalpoint p ∈ M , the Gaussian curvature of the surface at the criticalpoint p is the determinant of the Hessian of the function, i.e., thedeterminant of the two-by-two matrix of its second derivatives.

The implicit function theorem allows us to view any point of a reg-ular surface, as such a critical point, after a suitable rotation in R3. Wehave thus given the simplest possible definition of Gaussian curvatureat any point of a regular surface.

Remark 10.5.4. The appeal if this definition is that it allows oneimmediately to grasp the basic distinction between negative versus

10.5. INTRO TO THEOREMA EGREGIUM; RIEMANN’S FORMULA 125

positive curvature, in terms of the dichotomy “saddle point versuscup/cap”.

It will be more convenient to use an alternative definition in termsof the Weingarten map, which is readily shown to be equivalent to thedefinition in terms of the Hessian.

(1) (u1, u2) are coordinates in R2.(2) x = x(u1, u2) : R2 → R3 is a regular parametrized surface (i.e.,

the vector valued function x has Jacobian of rank 2 at everypoint).

(3) We have the partial derivatives xi =∂∂ui

(x), where i = 1, 2.Thus, vectors x1 and x2 form a basis for the tangent plane atevery point.

(4) We let xij =∂2x

∂ui∂uj∈ R3.

(5) 〈 , 〉 is the Euclidean inner product.(6) We let n = n(u1, u2) be a unit normal to the surface at the

point x(u1, u2), so that 〈n, xi〉 = 0.(7) The first fundamental form (gij) is given in coordinates by gij =

〈xi, xj〉.(8) The second fundamental form (Lij) is given in coordinates

by Lij = 〈n, xij〉.(9) The symbols Γkij are uniquely defined by the decomposition

xij = Γkijxk + Lijn

= Γ1ijx1 + Γ2

ijx2 + Lijn,(10.5.3)

where the repeated (upper and lower) index k implies summa-tion, in accordance with the Einstein summation convention.

(10) The Weingarten map (Lij) is an endomorphism of the tangentplane Rx1 +Rx2. It is uniquely defined by the decomposition

nj = Lijxi

= L1jx1 + L2

jx2,

where nj =∂∂uj

(n).

(11) We have the relation Lij = −Lki gjk.(12) We use the notation Γkij,ℓ for the ℓ-th partial derivative of the

symbol Γkij.(13) We will denote by square brackets [ ] the antisymmetrisa-

tion over the pair of indices found in between the brackets,e.g. a[ij] =

12(aij − aji).

(14) We have g[ij] = 0, L[ij] = 0, Γk[ij] = 0, and x[ij] = 0.


10.6. An identity involving the Γs and the Ls

Let x(u1, u2) be a regular parametrisation of a surface in R3.

Lemma 10.6.1. The third partial derivatives of x satisfy

xijk = (xij)k = (xik)j.

Proof. This is immediate from the equality of mixed partials of xi.�

The following formula will be used in the proof of the theoremaegregium of Gauss.

Proposition 10.6.2. On a surface M , we have the relation

Γqi[j,k] + Γmi[j Γqk]m = −Li[jLqk]

for each set of indices i, j, k, q (with, as usual, an implied summationover the index m).

Proof. We leave out the underlines for simplicity. Consider thethird partial derivative xijk =

∂3x∂ui∂uj∂uk

. Let us calculate its tangentialcomponent relative to the basis (x1, x2, n) for R3. Recall that nk =Lpkxp and xjk = Γℓjkxℓ + Ljkn. Thus, we have

(xij)k =(Γmijxm + Lijn

)k

= Γmij,kxm + Γmijxmk + Lijnk + Lij,kn

= Γmij,kxm + Γmij(Γpmkxp + Lmkn

)+ Lij (L

pkxp) + Lij,kn.

Grouping together the tangential terms, we obtain

(xij)k = Γmij,kxm + Γmij(Γpmkxp

)+ Lij (L

pkxp) + (. . .)n

=(Γqij,k + ΓmijΓ

qmk + LijL

qk

)xq + (. . .)n

=(Γqij,k + ΓmijΓ

qkm + LijL

qk

)xq + (. . .)n,

since the symbols Γqkm are symmetric in the two subscripts.By Lemma 10.6.1, the symmetry in j, k (equality of mixed partials)

implies the following identity: xi[jk] = 0. Therefore

0 = xi[jk]

= (xi[j)k]

=(Γqi[j,k] + Γmi[jΓ

qk]m + Li[jL

qk]

)xq + (. . .)n.

Since (x1, x2, n) is linearly independent, it follows that for each q = 1, 2,we have Γqi[j,k] + Γmi[jΓ

qk]m + Li[jL

qk] = 0. �

10.7. THE THEOREMA EGREGIUM OF GAUSS 127

10.7. The theorema egregium of Gauss

Recall from Section 9.10 that we have

K = det(Lij) = 2L1[1L

22] = − 2

g11L1[1L

22]. (10.7.1)

Theorem 10.7.1 (Theorema egregium). The Gaussian curvaturefunction K = K(u1, u2) of a surface can be expressed in terms of thecoefficients of the first fundamental form alone (and their first andsecond derivatives) as follows:

K =2

g11

(Γ21[1,2] + Γj1[1Γ

22]j

), (10.7.2)

where the symbols Γkij can be expressed in terms of the derivatives of gijbe the formula Γkij =


ℓk, where (gij) is the inversematrix of (gij).

In other words, we have

K =1

g11

(Γ211,2 − Γ2

12,1 + Γ111Γ

221 − Γ1

12Γ211 + Γ2

11Γ222 − Γ2

12Γ212

)(10.7.3)

Corollary 10.7.2. If the first fundamental form is diagonal, then

K = g11(Γ211,2 − Γ2

12,1 + Γ111Γ

221 − Γ1

12Γ211 + Γ2

11Γ222 − Γ2

12Γ212

)

Proof. If the matrix of the first fundamental form is diagonal thenone can make the substitution g11 = 1

g11in (10.7.3). �

Proof of theorema egregium. We present a streamlined version ofdo Carmo’s proof [Ca76, p. 233]. The proof is in 3 steps.

(1) We express the third partial derivative xijk in terms of both theΓ’s (intrinsic information) and the L’s (extrinsic information).

(2) The equality of mixed partials yields an identification of a suit-able expression in terms of the Γ’s, with a certain combinationof the L’s.

(3) The combination of the L’s is expressed in terms of Gaussiancurvature.

The first two steps were carried out in Proposition 10.6.2. Wechoose the valus i = j = 1 and k = q = 2 for the indices. ApplyingTheorem 9.10.1(c) (namely, identity (10.7.1)) we obtain

Γ21[1,2] + Γm1[1Γ

22]m = −L1[1L

22]

= g1iLi[1L

22]

= g11L1[1L

22]


since the term L2[1L

22] = 0 vanishes. This yields the desired formula

for K and completes the proof of the theorema egregium. �

10.8. Signed curvature as θ′(s)

Part of the motivation for introducing signed curvatures of planecurves (a refinement of the ordinary curvature of curves) is that theclassical definition of Gaussian curvature is in terms of the product ofsigned curvatures. Signed curvatures are defined below.

We identify R2 with C so that a vector in the plane can be written asa complex number. Let s be an arclength parameter along a curve α(s)with tangent vector v(s) = α′(s). Since v is of unit norm we can writeit as v = eiθ.

Definition 10.8.1 (function theta). The angle θ(s) along the curveα(s) with tangent vector v(s) = α′(s) is defined in one of the followingtwo equivalent ways:

(1) We write v(s) = eiθ(s), where the angle θ(s) is measured coun-terclockwise, from the positive ray of the x-axis, to the vec-tor v(s).

(2) Using a suitable branch of the complex logarithm, we can alsoexpress θ(s) as follows: θ(s) = 1

ilog v(s) = −i log v(s).

Here log is a suitable branch of the inverse of the complex expo-nential function, not to be confused with the real function ln.

Lemma 10.8.2. If α(s) = (x(s), y(s)) then dxds

= cos θ and dyds

=sin θ.

Proof. By definition we have v(s) = dxds

+ i dyds. Since v(s) = eiθ =

cos θ + i sin θ, we obtain the formulas of the lemma. �

Lemma 10.8.3. We have the relation ddθeiθ = ieiθ, where θ(s) is the

angle of Definition 10.8.1.

This was proved in complex functions. We are now ready to definethe signed curvature.

Definition 10.8.4. The signed curvature function kα of the planecurve α is

kα(s) =dθ

ds. (10.8.1)

This is discussed in more detail in Section 11.6. We have the follow-ing relation between the signed curvature and the ordinary curvatureof a curve as defined in Section 4.2.

Theorem 10.8.5. We have∣∣kα∣∣ = kα.

10.8. SIGNED CURVATURE AS θ′(s) 129

Proof. We differentiate v(s) = eiθ(s) by chain rule to obtain

dv

ds= ieiθ(s)

dθ

ds,

and therefore

|kα(s)| =∣∣∣∣dθ

ds

∣∣∣∣ =∣∣∣∣dv

ds

∣∣∣∣ =∣∣∣∣d2α

ds2

∣∣∣∣ = kα(s)

as required. �

CHAPTER 11

Signed curvature of curves; total curvature

11.1. Signed curvature with respect to arbitrary parameter


(1) α(s) an arclength parametrisation of a plane curve;(2) kα(s) = |α′′(s)| is the curvature of the (see formula (4.2.1);(3) θ(s) is the angle defined so that α′(s) = eiθ(s) in the plane;

(4) kα(s) =dθds

is the signed curvature (see Section 10.8).

The signed curvature of a plane curve can be expressed in terms ofan arbitrary parameter t of a regular parametrisation, as follows. Weuse dots (Newton’s notation) for derivatives with respect to t.

Theorem 11.1.1. Let α(t) = (x(t), y(t)) be an arbitrary regularparametrisation (not necessarily arclength) of a plane curve. Thensigned curvature satisfies the following equivalent formulas:

kα(t) =

xy−yx(x2+y2)3/2

det(α α)

|α|3 for the 2 by 2 matrix(α α).

(11.1.1)

Proof. With respect to an arbitrary parameter t, the componentsof α(t) are x(t) and y(t). Hence we have tan θ = y

xor equivalently

x sin θ = y cos θ. (11.1.2)

Differentiating (11.1.2) with respect to t, we obtain by product ruleand chain rule

x sin θ + x cos θdθ

dt= y cos θ − y sin θ

dθ

dt. (11.1.3)

Grouping together terms containing dθdt, we obtain

dθ

dt(x cos θ + y sin θ) = −x sin θ + y cos θ. (11.1.4)

The arclength parameter s is defined by s(t) =∫ t0|α| so that ds

dt= |α|.

Multiplying (11.1.4) by dsdt, we obtain

dθ

dt

(x cos θ

ds

dt+ y sin θ

ds

dt

)= −x sin θds

dt+ y cos θ

ds

dt.

131

132 11. SIGNED CURVATURE OF CURVES; TOTAL CURVATURE

Recall that by Lemma 10.8.2 we have{cos θ = dx

ds,

sin θ = dyds.

Therefore by chain rule, we obtain dθdt(x2 + y2) = −xy + yx so that

dθ

dt=xy − xy

x2 + y2.

Furthermore, dθdt

= dθdsdsdt

and therefore dθds

= xy−xy( ds

dt )3 since ds

dt= |α(t)| =

|dαdt|. Thus

dθ

ds=xy − yx

|α|3and by (10.8.1) we obtain the desired formulas (11.1.1) for the signedcurvature. �

Example 11.1.2. Calculate the curvature of the graph of y = f(x)at a critical point x = c using formula (11.1.1).

We will return to the study of signed curvature in Section 11.6.

11.2. Jordan curves; global geometry; Gauss map

In this section, we begin the study of the global geometry of planecurves. The curvature invariants we have studied thus far are localinvariants. The global geometry of surfaces will be studied in Sec-tion 12.4.

Definition 11.2.1. A Jordan curve in the Euclidean plane R2 =C is a non-selfintersecting closed curve that can be represented by acontinuous injective map α from the circle S1, or α : S1 → R2.

Theorem 11.2.2 (Jordan curve theorem). A Jordan curve sepa-rates the plane into two open regions: a bounded region and an un-bounded region.

The bounded region is called the interior region. We will only dealwith smooth (i.e., infinitely differentiable) regular maps α.

Consider a smooth Jordan curve J ⊆ C of length L. By Theo-rem 5.1.1 there is an arclength parametrisation α(s) of J , where

(1) s ∈ [0, L];(2) α(0) = α(L);(3) α′(0) = α′(L);(4) the tangent vector v(s) = α′(s) satisfies v(s) ∈ S1 ⊆ C

where S1 is the unit circle.

11.3. CONVEX JORDAN CURVES; ORIENTATION 133

This can be achieved by translating the initial point of the vec-tor v(s) to the origin. The normal vector n(s) = iv(s) defines a map Gto the unit circle called the Gauss map, as follows.

Definition 11.2.3. Assume a Jordan curve J ⊆ C is smooth. TheGauss map G of J of the map

G : J → S1, α(s) 7→ n(s) = iv(s) (11.2.1)

for each point α(s) ∈ J , where v(s) = α′(s).

Remark 11.2.4. The Gauss map is usually defined (as we did)using the normal vector n(s) (see Remark 11.5.5) though one coulduse v(s) as well.

Lemma 11.2.5. If the curve J is a circle of radius r centered at theorigin, then the Gauss map is the map G = 1

rIdJ , i.e., a multiple of

the identity map of the circle.

11.3. Convex Jordan curves; orientation

We begin the study of the global geometry of curves and surfaces.Recall the following set-theoretic notions.

(1) The set-theoretic complement of a subset B ⊆ A is denotedA \B.

(2) Any line ℓ ⊆ R2 divides the plane into two open halfplanes,namely the two connected components1 of the set comple-ment R2 \ ℓ.

Definition 11.3.1. A smooth Jordan curve J ⊆ R2 is called strictlyconvex 2 if one of the following two equivalent conditions is satisfied:

(1) the interior of each segment joining a pair of points of J iscontained in the interior region3 of J ;

(2) consider the tangent line Tp to J at a point p ∈ J ; then thecomplement J \ {p} lies entirely in one of the open halfplanesof R2 \ Tp defined by the tangent line, for all p ∈ J .

Recall that v(s) = α′(s) = eiθ(s) where θ ∈ [0, L].

Definition 11.3.2. A counterclockwise parametrisation (orienta-tion) of a convex Jordan curve J is the orientation defined by choosingthe arclength parameter so that the angle θ(s) increasing on [0, L],while a clockwise parametrisation (orientation) corresponds to θ(s) de-creasing.

1rechivei kshirut2kamur with kuf3Tchum pnimi


11.4. Properties of Gauss map

In Section 11.2 we defined the Gauss map G : J → S1 of a Jordancurve J ⊆ C in terms of the normal vector n(s).

Theorem 11.4.1. Assume that a smooth regular Jordan curve Jis strictly convex. Then the Gauss map G : J → S1 of (11.2.1) isone-to-one and onto.

Proof of the “onto” part. To fix ideas we will assume that Jis parametrized counterclockwise. We will work instead with the map

v : J → S1

defined by the tangent vector v. This is legitimate since the tangentvector only differs from the normal vector by a 90◦ rotation.

Step 1. Let α(s) be an arclength parametrisation of J . First weconsider the case of “horizontal” vectors v(s) in the (x, y)-plane. Theseoccur at points of J ⊆ C with maximal and minimal imaginary part,i.e., the y-coordinate.

Step 2. We think of the y-coordinate as defining a height functionon the curve. By applying Rolle’s theorem to the height function y(s),we obtain a minimum smin and a maximum smax. The points α(smin)and α(smax) have “opposite” horizontal tangent vectors. Such pointscorrespond to the values θ = 0 and θ = π. We have

G(α(smin)) = ei0, G(α(smax)) = eiπ.

This exhibits the points ei0 ∈ S1 and eiπ ∈ S1 as images of the Gaussmap.

Step 3. To treat the general case, the idea is to use an analog ofthe height function which is, up to sign, the distance to the line SpanRvspanned by the vector v = eiθ. Let us show how one obtains the pairof opposite tangent vectors

v = eiθ and − v = ei(θ+π). (11.4.1)

Consider the vector

nθ = ei(θ+π2) (11.4.2)

normal to v. Consider the function h(s) (h for “height”) given by thescalar product

h(s) = 〈α(s), nθ〉 (11.4.3)

The function h is analogous to the y-coordinate in Step 2 above.

11.5. TOTAL CURVATURE OF A CONVEX JORDAN CURVE 135

Step 4. We seek the extrema of the function h of (11.4.3). At acritical point s0 of the function h, we have by Fermat’s theorem,

d

ds

∣∣∣∣s=s0

h(s) =

⟨dα

ds, nθ

⟩= 0.

where nθ is the normal vector of (11.4.2). Hence the tangent vec-tor v(s0) = α′(s0) at each critical point s0 of h is parallel to the vec-tor v of (11.4.1). The minimum and the maximum of h give the tan-gents v = eiθ and −v = ei(θ+π), which are therefore both in the imageof the map G.

Step 5. As v ranges over S1, we thus obtain the points on thecurve where the tangent vector is parallel to v, provng surjectivity. �

Proof of the “one-to-one” part. We would like to show thatthe Gauss map (11.2.1) is one-to-one. As before, we will work withtangent vectors in place of the normal vectors.

Step 1. We argue by contradiction. Suppose on the contrarythat two distinct points p ∈ J and q ∈ J have identical tangent vec-tors v(s) = eiθ. Then the tangent lines Tp and Tq to J at p and q areparallel.

Step 2. By definition of convexity, the curve J lies entirely on oneside of each of the tangent lines Tp and Tq. Since the lines are parallel,there are two possibilities:

(1) the tangent lines coincide: Tp = Tq.(2) The curve lies in the strip between the two tangent lines.

However, in the second case the tangent vectors at p and q have oppositedirections v abd −v. It remains to treat the case Tp = Tq.

Step 3. Thus we can assume that both p and q must lie onthe common line Tp = Tq. Therefore we obtain a straight line seg-ment [p, q] ⊆ J . This contradicts the hypothesis that the curve Jis strictly convex. The contradiction proves that the map is one-to-one. �

11.5. Total curvature of a convex Jordan curve

The result on the total curvature4 of a Jordan curve is of interest inits own right. Furthermore, the result serves to motivate an analogousstatement of the Gauss–Bonnet theorem for surfaces in Section 12.7.

4Akmumiyut kolelet


Definition 11.5.1. The total curvature Tot(C) of a curve C witharclength parametrisation α(s) : [a, b] → R2 is the integral

Tot(C) =

∫ b

a

kα(s)ds. (11.5.1)

Lemma 11.5.2. The total curvature of a circle Cr (of radius r) is 2πand therefore indepedent of r.

Proof. Consider an arclength parametrisation

α(s) : [0, 2πr] → R2

of the circle Cr given by the usual trigonometric functions. Then

Tot(Cr) =

∫ 2πr

0

kα(s)ds =

∫ 2πr

0

1

rds =

2πr

r= 2π.

�

Definition 11.5.3. If C is a twice differentiable smooth closedcurve parametrized by α : [a, b] → R2, so that in particular α(a) = α(b)and α′(a) = α′(b), we will write the integral (11.5.1) using the notationof a contour integral

Tot(C) =

∮

C

kα(s)ds. (11.5.2)

Theorem 11.5.4. Let C be a strictly convex smooth Jordan curvewith arclength parametrisation α(s). Then the total curvature of C isTot(C) =

∮Ckα(s)ds = 2π.

Proof. Let α(s) be a unit speed parametrisation chosen so that α(0)is the lowest point (i.e., y is minimal) of the curve, and assume the curveis parametrized counterclockwise. Let

v(s) = α′(s) ∈ C.

Then v(0) = ei0 = 1 and therefore θ(0) = 0. The function θ = θ(s)is monotone increasing from 0 to 2π by Theorem 11.4.1. Applying thechange of variable formula for integration, we obtain

Tot(C) =

∮

C

kα(s)ds =

∮

C

∣∣∣∣dv

ds

∣∣∣∣ ds =∮

C

dθ

dsds =

∫ 2π

0

dθ = 2π,


Remark 11.5.5. We can also consider the normal vector n(s) tothe curve. The normal vector satisfies n(s) = ei(θ+

π2) = iei(θ) and |dn

ds| =

11.6. ROTATION INDEX OF A CLOSED CURVE IN THE PLANE 137

|dvds| = dθ

ds. Therefore we can also calculate the total curvature as follows:∮

C

kα(s)ds =

∮

C

∣∣∣∣dn

ds

∣∣∣∣ ds =∮

C

dθ

dsds =

∫ 2π

0

dθ = 2π,

with n(s) in place of v(s). For surfaces, an analogous Gauss map isdefined by means of the normal vector.

Applying Theorem 11.5.4 we obtain the following.Nondegeneracy of conic sections was discussed in Definition 2.6.5

and the main text there.

Corollary 11.5.6. Consider a closed convex curve given by anellipse E ⊆ R2 defined by a quadratic equation

ax2 + 2bxy + cy2 + dx+ ey + f = 0, ac− b2 > 0,

and assume that the ellipse is nondegenerate. Then

Tot(E) = 2π.

Namely 2π is the total curvature of the ellipse E.

Example 11.5.7. The hyperbola H defined by λ1x2 + λ2y

2 = k isnot a closed curve. In the special case λ1 = −λ2 (see Definition 2.6.2)when the asymptotes of H are orthogonal, the image of θ is preciselyhalf the circle. Therefore we can define the total curvature of this non-closed curve by a similar integral, and a similar integration argumentshows that the total curvature is π (rather than 2π).

Remark 11.5.8 (Non-convex curves). A theorem similar to The-orem 11.5.4 in fact holds for an arbitrary regular Jordan curve (eventhough in general θ(s) will not be a monotone function) provided weuse signed curvature. We have dealt only with the convex case in orderto simplify the topological considerations. See further in Section 11.6.

Remark 11.5.9. A similar calculation will yield the Gauss–Bonnettheorem for convex surfaces in Section 12.7.

11.6. Rotation index of a closed curve in the plane

In this section we will deal with arbitrary closed curves (not nec-essarily convex). Let α(s) be an arclength parametrisation of a closedcurve in the plane. We assume that the curve is parametrized counter-clockwise. Let v(s) = α′(s). As in Section 11.2, we have the followingresult.

The following result applies to closed curves that are not necessarilysimple (i.e., non-self-intersecting).


Figure 11.6.1. Three points on a curve with thesame tangent direction, but the middle one correspondsto θ′(s) < 0.

Theorem 11.6.1. For a smooth curve α(s) in the plane, a contin-uous single-valued branch of θ(s), s ∈ [0, L], can be chosen where θ(s)is the angle measured counterclockwise from the positive x-axis to thetangent vector v(s) = α′(s).

Remark 11.6.2. If the closed curve is not convex, the function θ(s)will not be monotone and at certain points its derivative may takenegative values:

θ′(s) < 0.

Once we have chosen a continuous branch of θ, we can definethe signed curvature kα of the parametrized closed curve as in Def-inition 10.8.4 by setting kα(s) = dθ

dswhere θ = 1

ilog v, or equiva-

lently v(s) = α′(s) = eiθ(s).Consider a regular closed plane curve of length L with an arclength

parametrisation α(s) with α′(s) = eiθ(s). By Theorem 11.6.1, a con-tinuous branch of θ(s) can be chosen even if the curve is not simple.Such a branch is a map θ : [0, L] → R. The values θ(L) and θ(0) mustagree up to a multiple of 2π since they define the same tangent vector.

11.8. GAUSS–BONNET THEOREM FOR A PLANE DOMAIN 139

Therefore the difference θ(L) − θ(0) is necessarily an integer multipleof 2π.

Definition 11.6.3. The rotation index ια of a closed unit speed

plane curve α(s) is the integer ια = θ(L)−θ(0)2π

.

Theorem 11.6.4. The rotation index of a smooth Jordan curve Jis ιJ = ±1 (namely, 1 for the counterclockwise orientation and −1 forthe clockwise orientation).

For a proof see Millman & Parker [MP77, p. 55]. We will analyzethe rotation index further in Section 13.1.

11.7. Total signed curvature of Jordan curve

We have the following generalisation of Theorem 11.5.4 on the totalcurvature.

Definition 11.7.1. The total signed curvature of a smooth closedcurve C ⊆ R2 of length L with arclength parametrisation α(s) fors ∈ [0, L] is defined to be

Tot(C) =

∫ L

0

kα(s)ds.

We obtain the following generalisation of Theorem 11.5.4. As usualwe assume that Jordan curves are parametrized counterclockwise.

Theorem 11.7.2. Suppose C is a Jordan curve oriented counter-clockwise. Then the total signed curvature of C is

Tot(C) = 2π. (11.7.1)

Proof. We exploit a continuous branch θ(s) as in Theorem 11.5.4,but without the absolute value signs on the derivative of θ(s). Applyingthe change of variable formula for integration, we obtain

Tot(C) =

∮

C

kα(s)ds =

∮

C

dθ

dsds =

∫ θ(0)+2π

θ(0)

dθ = 2π,

where the upper limit of integration is θ(0)+ 2π by Theorem 11.6.4 onthe rotation index. �

11.8. Gauss–Bonnet theorem for a plane domain

There is a generalisation (to plane domains) of the formula (11.7.1)for the total signed curvature of a Jordan curve.


Definition 11.8.1. The Euler characteristic χ(D) of a surface D,defined via a triangulation, is χ(D) = V −E+F (vertices minus edgesplus faces).

Theorem 11.8.2. Consider a plane domain D ⊆ R2 with (possiblyseveral) smooth bundary components. Assume that each of the compo-nents of ∂D is oriented in a way compatible with the standard orienta-tion in D, and let k be the signed curvature. Then

∫∂Dk = 2πχ(D).

We mention some examples.

(1) If D is a disk then χ(D) = 1 and total curvature of the bound-ary of D is 2π.

(2) If D is an annulus then χ(D) = 0 and total curvature of theboundary of D is 0.

(3) If D ⊆ R2 has two holes (in addition to the outside boundary)then then χ(D) = −1 and total curvature of the boundaryis −2π.

A more general Gauss–Bonnet theorem for surfaces is discussed inSections 12.8 and 12.11.

11.9. Gaussian curvature as product of signed curvatures

We will exploit signed curvature defined in Section 10.8 to expressGaussian curvature as a product of signed curvatures of planar curves.

We originally defined the Gaussian curvature K = K(u1, u2) asthe determinant of the Weingarten map Wp at p = x(u1, u2) ∈ M .We also saw that the absolute value |K| can be represented as theproduct of the curvatures kβ1 and kβ2 of geodesics in the direction ofthe eigenvectors vi of Wp, namely |K| = kβ1kβ2 .

One can replace the geodesics by plane curves obtained by inter-sectingM with the planes spanned by n and each of the eigenvectors vi,as already mentioned in Definition 9.7.1, item (1). Then for signed cur-

vatures one has the sharper formula K = kβ1 kβ2 as in Theorem 11.9.2below. In this spirit, some textbooks (following Gauss himself) definethe Gaussian curvature not as the determinant of the Weingarten mapbut rather as the product of the signed curvatures kα of such a pair ofplane curves.

Definition 11.9.1. We consider the following data.

(1) M ⊆ R3 is a surface.(2) p ∈M .(3) v1, v2 are orthonormal eigenvectors of Wp : Tp → Tp.(4) The plane Ei = Span(vi, n) ⊆ R3 is spanned by eigenvector vi

and n for i = 1, 2.

11.11. MEAN VERSUS GAUSSIAN CURVATURE 141

(5) The plane curve βi =M ∩Ei is the intersection of the surfaceand the plane.

(6) Choose unit speed parametrisations βi(s) such that β′i(0) = vi.

Choose a unit normal n to the surface M at p ∈M . We can definethe signed curvature ki (relative to n) of geodesics βi(s) by setting

ki = 〈β′′i , n〉.

Theorem 11.9.2. The Gaussian curvature Kp of M ⊆ R3 at psatisfies

Kp = k1k2 (11.9.1)

where ki, i = 1, 2 is the signed curvature of the plane curves β1 and β2.

11.10. Connection to the Hessian

To explain formula (11.9.1) in terms of the Hessian, we can assumewithout loss of generality that M is the graph of a function f(x, y)vanishing at the origin, with a critical point at the origin.

Suppose the eigendirections of the Hessian of f are precisely the x-axis and the y-axis. Then the Gaussian curvature at the origin is theproduct of signed curvatures of the curves obtained as the graphs ofthe restrictions of f respectively to the x-axis and the y-axis.

If the eigenspaces are not the axes, we adjust the coordinates by asuitable rotation of R3 so that the x and y axes become the eigenspaces,while the z-axis is the direction of the normal. Note that Gaussiancurvature is unchanged under such a rotation.

(1) The curve β1 ⊆ E1 in the (x, z) plane is the graph of z =ax2 + o(x2);

(2) the curve β2 ⊆ E2 in the (y, z) plane is the graph of z =by2 + o(y2).

With respect to the normal n = e3, the curves have signed curva-ture respectively 2a and 2b at the origin. The Hessian is the diagonal

martix Hf =

(2a 00 2b

). Therefore K = 4ab as required.

11.11. Mean versus Gaussian curvature

Recall that by Gauss’ theorema egregium the Gaussian curvaturecan be expressed in terms of the metric coefficients alone.

Theorem 11.11.1. Unlike Gaussian curvature K, the mean curva-ture H = 1

2Lii cannot be expressed in terms of the metric coefficients gij

and their derivatives.


Weingartenmap (Lij)

Gaussian cur-vature K

mean cur-vature H

plane

(0 00 0

)0 0

cylinder

(1 00 0

)0 1

2

invarianceof curvature

yes no

Table 11.11.1. Plane and cylinder have the same intrinsicgeometry (K), but different extrinsic geometries (H)

Namely, the plane and the cylinder have parametrisations withidentical (gij), but with different mean curvature, cf. Table 11.11.1.To summarize, Gaussian curvature is an intrinsic invariant, while meancurvature an extrinsic invariant, of the surface.

11.11.1. Algebraic degree. This material is optional. Let α(s) be aparametrisation of a smooth closed curve C. Consider an arbitrary smoothmap (not necessarily the Gauss map) α(s) 7→ eiθ(s) from C to the circle.The algebraic degree of the map at a point z ∈ S1 is defined to be thesum

∑θ−1(z) sign

(dθds

), where the summation is over all points in the inverse

image of z. Here by Sard’s theorem z can be chosen in such a way that theinverse image is finite so that the sum is well-defined.

Theorem 11.11.2. For Jordan curves (i.e., embedded curves), the alge-braic degree of the Gauss map is 1 if oriented counterclockwise and −1 iforientated counterclockwise.

For nonconvex Jordan curves, the Gauss map to the circle will not beone-to-one, but will still have an algebraic degree one. This phenomenonhas an analogue for embedded surfaces in R3 where the algebraic degree isproportional to the Euler characteristic.

Example 11.11.3. If the curve is not simple, the degree may be differentfrom ±1. Thus, the map defined by z 7→ zn restricted to the unit circle givesa map eiθ 7→ einθ of algebraic degree n. The preimage of 1 = ei0 consists

precisely of the nth roots of unity e2πikn , k = 0, 1, 2, . . . , n − 1. Altogether

there are n of them and therefore the algebraic degree is n.

Connected components of curves. This material is optional. Untilnow we have only considered connected curves. A curve may in generalhave several connected components (rechivei k’shirut). Two points p, q ona curve C ⊆ R2 are said to lie in the same connected component if there

11.11. MEAN VERSUS GAUSSIAN CURVATURE 143

exists a continuous map h : [0, 1] → C such that h(0) = p and h(1) = q.This defines an equivalence relation on the curve C, and decomposes itas a disjoint union C =

⊔iCi. The set of connected components of C is

denoted π0(C). The number of connected components is denoted |π0(C)|.Example 11.11.4. Let F (x, y) = (x2 + y2 − 1)((x− 10)2 + y2 − 1), and

let CF be the curve defined by F (x, y) = 0. Then CF is the union of a pair ofdisjoint circles. Therefore it has two connected components: |π0(CF )| = 2.

The total curvature can be similarly defined for a non-connected curve,by summing the integrals over each connected components.

Definition 11.11.5. The total curvature of a curve C =⊔iCi is Tot(C) =∑

iTot(Ci).

We have the following generalisation of the theorem on total curvatureof a curve.

Theorem 11.11.6. If each connected component of a curve C is a Jordancurve parametrized counterclockwise, then the signed total curvature of C

is Tot(C) = 2π |π0(C)|.Proof. We apply the previous theorem to each connected component,

and sum the resulting total curvatures. �

CHAPTER 12

Laplace–Beltrami, Gauss–Bonnet

12.1. The Laplace–Beltrami operator

LetM be a surface. Recall that a (u1, u2)-chart in which the metricon M becomes conformal (see Definition 7.7.1)1 to the standard flatmetric, is referred to as isothermal coordinates. The existence of thelatter is proved in [Bes87].

Consider a metric onM of the form λ(u1, u2)δij, where λ > 0. Then(u1, u2) are isothermal coordinates.

Definition 12.1.1. The Laplace–Beltrami operator of the met-ric λ(u1, u2)δij is the operator

∆LB =1

λ

(∂2

∂(u1)2+

∂2

∂(u2)2

).

Remark 12.1.2. The notation means that when we apply the op-erator ∆LB to a function h = h(u1, u2), we obtain

∆LB(h) =1

λ

(∂2h

∂(u1)2+

∂2h

∂(u2)2

).

In more readable form, for a metric gij = λ(x, y)δij and a func-tion h = h(x, y), we have

∆LB(h) =1

λ

(∂2h

∂x2+∂2h

∂y2

).

In other words ∆LBh = 1λ∆0h where ∆0 is the flat Laplacian with

respect to coordinates (x, y).

12.2. Laplace-Beltrami formula for Gaussian curvature

Let ln x be the natural logarithm so that ddx

ln x = 1x.

Theorem 12.2.1. Given a metric in isothermal coordinates withmetric coefficients gij = λ(u1, u2)δij, its Gaussian curvature is minus

1See also Definition 16.10.1

145

146 12. LAPLACE–BELTRAMI, GAUSS–BONNET

Γ1ij j = 1 j = 2

i = 1 µ1 µ2

i = 2 µ2 −µ1

Γ2ij j = 1 j = 2

i = 1 −µ2 µ1

i = 2 µ1 µ2

Table 12.2.1. Symbols Γkij of a metric e2µ(u1,u2)δij

half the Laplace-Beltrami operator applied to the ln of the conformalfactor λ:

K = −1

2∆LB(lnλ). (12.2.1)

Proof. The Gamma symbols were computed in Section 7.7.

Step 1. We write λ = e2µ and tabulate the Γ in Table 12.2.1. Wehave from Table 12.2.1 (based on Table 7.7.1):

2Γ21[1,2] = Γ2

11,2 − Γ212,1 = −µ22 − µ11.

Step 2. The ΓΓ term in the expression (10.7.2) for the Gaussiancurvature vanishes:

2Γj1[1Γ22]j = 2Γ1

1[1Γ22]1 + 2Γ2

1[1Γ22]2

= Γ111Γ

221 − Γ1

12Γ211 + Γ2

11Γ222 − Γ2

12Γ212

= µ1µ1 − µ2(−µ2) + (−µ2)µ2 − µ1µ1

= 0.

Step 3. Since the ΓΓ term vanishes by Step 2, from formula (10.7.2)we have

K =2

λΓ21[1,2] = −1

λ(µ11 + µ22) = −∆LBµ.

Meanwhile, ∆LB lnλ = ∆LB(2µ) = 2∆LB(µ), proving the result. �

Setting λ = f 2, we can restate the theorem as follows.

Corollary 12.2.2. Given a metric in isothermal coordinates withmetric coefficients gij = f 2(u1, u2)δij, its Gaussian curvature is minusthe Laplace-Beltrami operator of the ln of the conformal factor f :

K = −∆LB(ln f). (12.2.2)

Either one of the formulas (10.7.2), (12.2.1), or (12.2.2) can serveas the intrinsic definition of Gaussian curvature, replacing the extrinsicdefinition (10.7.1); cf. Remark 10.5.

12.3. HYPERBOLIC METRIC IN THE UPPER HALF-PLANE 147

12.3. Hyperbolic metric in the upper half-plane

Let x = u1 and y = u2.

Definition 12.3.1. The upper half-plane is H2 = {(x, y) : y > 0}.Theorem 12.3.2. The metric with coefficients

gij =1

y2δij (12.3.1)

in the upper half-plane H2 has constant Gaussian curvature K = −1.

Proof. By Theorem 12.2.2, we have

K = −∆LB ln f = ∆LB ln y = y2(− 1

y2

)= −1,

as required. �

Definition 12.3.3. The metric (12.3.1) is called the hyperbolicmetric of the upper half-plane.

This example is also discussed in Section 16.20.

Theorem 12.3.4 (Geodesics in the upper half-plane). The differ-ential equations of a geodesic for the hyperbolic metric in the upperhalf-plane are

{y x′′ − 2x′y′ = 0 for k = 1

y y′′ + (x′)2 − (y′)2 = 0 for k = 2.

Proof. In Section 7.7 we computed the Gamma symbols of a met-ric relative to isothermal coordinates.

Step 1. We have Γ111 = λ1

2λ, Γ1

22 = −λ12λ

, and Γ112 = λ2

2λ. Since

λ(x, y) = y−2 for the hyperbolic metric we obtain{Γ111 = Γ1

22 = 0

Γ112 =

λ22λ

= −2y−3

2y−2 = −y2

y3= − 1

y.

(12.3.2)

Step 2. For k = 1, the differential equation of a geodesic, namely

αk′′+ Γkijα

i′αj′= 0,

becomes

α1′′ + 2Γ112α

1′α2′ = 0.

Since α1 = x and α2 = y we obtain x′′ + 2Γ112x

′y′ = 0 which usingformula (12.3.2) becomes x′′ − 2

yx′y′ = 0.


Step 3. Now let k = 2. We have

Γ211 = −λ2

2λ= 1

y

Γ222 = − 1

y

Γ212 =

λ12λ

= 0.

Therefore the geodesic equation for k = 2 is y′′+ 1yx′x′− 1

yy′y′ = 0. �

Example 12.3.5 (Vertical ray). The exponential function y = es

(while x is constant as a function of s) satisfies both equations of The-orem 12.3.4. Indeed, for a vertical line y = y(t), the second equationgives yy′′ − (y′)2 = 0 or (

y′

y

)′= 0.

Therefore y′

y= C, or dy

ds= Cy. Separating the variables we obtain

dyy= Cds. Integrating, we obtain ln y = Cs, or y = eCs. Therefore y =

eCs provides a constant speed parametrisation of a hyperbolic geodesicwhich traces out a vertical half-line in the upper half-plane.

Remark 12.3.6. The pseudosphere of Section 5.10 is an exampleof a surface of constant Gaussian curvature −1 embedded in Euclideanspace; see Section 13.3.

12.4. Area elements of the surface, orientability

Consider a surface M ⊆ R3 with local parametrisation x(u1, u2).We have √

det(gij) =∣∣x1 × x2

∣∣ ,where xi =

∂x∂ui

.2 We saw two cases of area elements of surfaces:

(1) the case of the area element rdrdθ in the plane in polar coor-dinates as in Section 8.3;

(2) the spherical element of area sinϕdθ dϕ,

More generally, we have the following definition, already mentionedin Definition 8.6.1.

Definition 12.4.1. The area element dAM of the surface M is

dAM =√det(gij) du

1du2 = |x1 × x2| du1du2 (12.4.1)

where the gij are the metric coefficients of the surface with respect tothe parametrisation x(u1, u2).

2Cf. the Binet–Cauchy identity 12.11.4.

12.5. GAUSS MAP FOR SURFACES IN R3 149

We have used the subscriptM so as to specify which surface we aredealing with.3

Definition 12.4.2. A surface M ⊆ R3 is orientable if it admits acontinuous unit normal vector field N = Np, defined at each point p ∈M .

The vector field N is a globally defined field along an orientablesurface. Note that the image vector N can be thought of as an elementof the unit sphere: Np ∈ S2 ⊆ R3. This observation leads to thedefinition of the Gauss map for surfaces in Section 12.5.

12.5. Gauss map for surfaces in R3

Let M ⊆ R3 be an orientable surface. Let N be a globally definednormal vector field along M .

Remark 12.5.1. Each point p ∈ M lies in a neighborhood witha suitable parametrisation x(u1, u2). At a point p = x(u1, u2), wehave a normal vector n(u1, u2), obtained by normalizing the vectorproduct x1 × x2, which coincides with the globally selected normal N .Thus, we have

n(u1, u2) = Nx(u1,u2).

Definition 12.5.2. The Gauss map of an orientable surface M ⊆R3, denoted

G : M → S2,

is the map sending each point p = x(u1, u2) to the normal vectorG(p) =n(u1, u2), where at every point we choose the normal that coincideswith the global vector field N at p.

Example 12.5.3. IfM = R2 is the xy-plane then the normal vectorat every point is e3 and therefore G(p) = e3 for all points p ∈M .

Example 12.5.4. If M = S2r is the sphere of radius r > 0 then

G(p) = 1rp for all p ∈M .

Recall that Wp : Tp → Tp is the Weingarten map.

Lemma 12.5.5. If K 6= 0 at p ∈ M then the tangent vectorsWp(x1),Wp(x2) ∈ Tp are linearly independent.

Proof. The coefficients of the matrix (Li j) the coordinates of∂n∂u1, ∂n∂u2

with respect to the basis (x1, x2). Hence independence of thevectors n1 and n2 is equivalent to the condition K = det(Li j) 6= 0. �

3This notion of area is discussed in more detail in Section 16.9.


Theorem 12.5.6. Choose a point p = x(a, b) ∈ M of the surface.If the Gaussian curvature is nonzero at p, then the map n(u1, u2) from(a neighborhood of) M to S2 produces a regular parametrisation of aspherical neighborhood of the point

G(p) = n(a, b) ∈ S2

on the sphere, where G is the Gauss map.

Proof. The parametrisation is given by the map G to the sphere.We have

∂

∂ui(G(p)) =

∂

∂ui(n(u1, u2)) =

∂n

∂ui= Wp(xi).

By Lemma 12.5.5, the two partial derivatives of the Gauss map G arelinearly independent. Hence the parametrisation defined by n(u1, u2)is regular in a neighborhood of p. �

12.6. Sphere S2 parametrized by Gauss map of M

The vector n(u1, u2) was originally defined as a normal to the origi-nal surface M ⊆ R3. We now with to think of it as giving a parametri-sation of an open neighborhood on the unit sphere via the Gauss mapof M .

Definition 12.6.1. AssumeKp 6= 0 at a point p ∈M , and considera neighborhood of the point G(p) in S2. Let

gαβ = nα · nβbe the metric coefficients of the regular parametrisation n(u1, u2) in

the neighborhood, where nα = ∂n(u1,u2)∂uα

for all α = 1, 2, β = 1, 2.

Now consider the area element dA (defined in Section 12.4) of theunit sphere.

Theorem 12.6.2. Consider a parametrisation n(u1, u2) of a neigh-borhood on the sphere S2 as in Theorem 12.5.6. Then the area ele-ment dAS2 of S2 can be expressed as

dAS2 =√

det(gαβ) du1du2 = |n1 × n2|du1du2.

Proof. This follows from the usual formula for the element of areafor surfaces, applied to the chosen parametrisation n(u1, u2) as definedin Theorem 12.5.6, in place of the standard parametrisation in sphericalcoordinates.4 �

4Cf. Binet–Cauchy in note 12.11.2.

12.7. COMPARISON OF TWO PARAMETRISATIONS 151

Remark 12.6.3. We have used the subscript S2 to distinguish thearea element dAS2 of the sphere S2 from the area element dAM of thesurfaceM as in (12.4.1). Note that we need to distinguish the two areaelements

dAS2 and dAM

because both will occur in the proof of the Gauss–Bonnet theorem.

12.7. Comparison of two parametrisations

When the Gaussian curvature is nonzero, the normal vector ofM ⊆R3 allows us to define local coordinates n(u1, u2) on the unit sphere S2

as in Section 12.5.

Proposition 12.7.1. Consider the following data:

(1) the metric coefficients (gij) of the parametrisation x(u1, u2) ofthe surface M , and

(2) the metric coefficients (gαβ) of the sphere S2 relative to theparametrisation n(u1, u2).

Then we have the identity√det(gαβ) = |K(u1, u2)|

√det(gij)

where K = K(u1, u2) is the Gaussian curvature of the surface M .

Proof. Let L = (Lij) be the matrix of the Weingarten map Wp

with respect to the basis (x1, x2). Here p = x(u1, u2). By definition ofcurvature we have K = det(L). Recall that the coefficients Lij of theWeingarten map are defined via the decomposition

nα = xiLiα. (12.7.1)

Consider the 3 × 2-matrices A = [x1 x2] and B = [n1 n2]. Thenformula (12.7.1) implies by definition of matrix multiplication that

B = AL.

Therefore the corresponding Gram matrices satisfy

Gram(n1, n2) = BtB = (AL)tAL = Lt(AtA)L = Lt Gram(x1, x2) L.

Applying the determinant, we obtain

det (Gram(n1, n2)) = det (Gram(x1, x2)) det2(L),

completing the proof of the lemma since det(L) = K, and the Grammatrix represents the first fundamental form. �


5

12.8. Gauss–Bonnet theorem for surfaces with K > 0

Both the Gauss–Bonnet theorem for surfaces and the theorem onthe total curvature of a plane curve (Theorem 11.5.4) assert that anintegral of curvature has topological significance. See also Section 11.7for a version of the theorem for plane domains. We will only treat theGauss–Bonnet theorem in the case K > 0.

Example 12.8.1. A typical example of a compact surface of posi-tive Gaussian curvature is an ellipsoid.

Theorem 12.8.2 (Special case of Gauss–Bonnet). Let M be anorientable compact surface in R3 of positive Gaussian curvature. Thenthe curvature integral satisfies

∫

M

Kp dAM = 4π,

where Kp is the Gaussian curvature of M at the point p.

Remark 12.8.3. Note that 4π = 2π χ(S2) where χ is the Eulercharacteristic (see Section 12.11).

To prove Theorem 12.8.2, we will use the existence of global coor-dinates (with singularities only at the north and south poles) on thesphere to write down a concrete version of the calculation. We recallthe following.

(1) x(u1, u2) is a regular parametrisation of a surface M ⊆ R3.(2) At every point ofM where the parametrisation x is defined, we

have the normal vector n(u1, u2) so that (x1, x2, n) is a basisfor R3.

(3) If M is an orientable surface, a continuous Gauss map

G : M → S2

is defined by the normal vector n(u1, u2) as in Section 12.5.(4) (θ, ϕ) are the usual spherical coordinates on the unit sphere.(5) IfKp 6= 0 then the Gauss map G :M → S2 is invertible near p.

5By the Cauchy-Binet formula, the desired identity is equivalent to the for-mula |nu1 ×nu2 | = |det(Li

j)| |xu1 ×xu2 |, immediate from the observation that a lin-ear map multiplies the area of parallelograms by its determinant. Namely, the Wein-garten map sends each vector xi to ni:Wp(xi) = ni ∈ Tp. A stronger identity is truethat is sensitive to the sign of the Gaussian curvature:W (u)×W (v) = det(W )(u×v)where (u, v) is any basis of the tangent plane; e.g., the basis (x1, x2).

12.9. PROOF OF GAUSS–BONNET THEOREM 153

Since the Gauss map is invertible, we can consider the inverse

G−1 : S2 →M. (12.8.1)

Definition 12.8.4. Let Iθ = [0, 2π] and Iϕ = [0, π]. We considerthe usual parametrisation σ = σ(θ, φ) : Iθ × Iϕ → S2 given by

σ(θ, ϕ) = (sinϕ cos θ, sinϕ sin θ, cosϕ) ∈ S2,

of the sphere as a surface of revolution (omitting the problematic poles),as in Section 5.5.

We also construct a special parametrisation of M based on theparametrisation σ of the sphere, as follows.

Definition 12.8.5. Let x(θ, ϕ) be the parametrisation of M givenby the composition x = G−1 ◦ σ using (12.8.1):

Iθ × Iϕσ−→ S2 G−1

−→M,

and denote by (gij) the corresponding metric coefficients.

12.9. Proof of Gauss–Bonnet Theorem

We can dispense with local coordinate charts and instead computethe Gauss–Bonnet integral relative to coordinates (θ, ϕ) as in Defi-nition 12.8.5 (because they only omit two points). Let (gij) be themetric coefficients of M with respect to the parametrisation of Defini-tion 12.8.5. Then the area element of M can be written as

dAM =√

det(gij) dθdϕ.

We will exploit the relation

K(θ, ϕ)√

det(gij) =√det(gαβ) (12.9.1)

from Section 12.7, where u1 = θ and u2 = ϕ. Here gαβ are the metriccoefficients of the standard metric on S2 with respect to parametrisa-tion defined by the Gauss map G ofM . We write the double integral

∫∫

as shorthand for the iterated integral∫ ϕ=πϕ=0

∫ θ=2π

θ=0. Recall that the area

element of M is

dAM =√

det(gij) dθdϕ.


We therefore obtain using (12.9.1):∫

M

K(θ, ϕ) dAM =

∫∫K(θ, ϕ)

√det(gij) dθdϕ

=

∫∫ √det(gαβ) dθdϕ

=

∫∫dAS2

= area(S2) = 4π,

as required.6

12.10. Gauss–Bonnet with boundary

There are various generalisations of the above theorem. One ofthem is the following version with boundary.

Theorem 12.10.1 (Version with boundary of Gauss–Bonnet theo-rem). Let M be a surface. Consider a geodesic triangle (homeomorphicto a disk) T ⊆M with angles α, β, γ. Then the integral of the Gaussiancurvature over T is the angular excess:∫

T

KdA = α + β + γ − π.

Corollary 12.10.2. The area of a spherical triangle with angles α,β, and γ is the angular excess α + β + γ − π.

Proof. We apply the local Gauss–Bonnet theorem and notice thatfor M = S2, we have K = 1 at every point. �

12.11. Euler characteristic and Gauss–Bonnet

Definition 12.11.1. Assume a surface M is partitioned into tri-angles. Then the Euler characteristic χ(M) of M is

χ(M) = V − E + F,

where V,E, F are respectively the numbers of vertices, edges, and faces(i.e., triangles) of M .

6Alternative proof. This material is optional. Here we present another proofof Gauss–Bonnet theorem. The convexity of the surface guarantees that the map nis one-to-one (compare with the proof of Theorem 11.4.1 on closed curves). Theintegrand K dAM in a coordinate chart (u1, u2) can be written as K(u1, u2) dAM .

By Proposition 12.7.1, we have K(u1, u2) dAM = K(u1, u2)√det(gij)du

1du2 =√det(gαβ)du

1du2 = dAS2 . Thus, the expression K dAM coincides with the areaelement dAS2 of the unit sphere S2 in every coordinate chart. Hence we canwrite

∫MK dAM =

∫S2 dAS2 = 4π, proving the theorem.

12.11. EULER CHARACTERISTIC AND GAUSS–BONNET 155

The Euler characteristic of a closed embedded surface in Euclidean3-space can be computed via the integral of the Gaussian curvature. Itcan be thought of as a generalisation of the rotation index of a planeclosed curve.

Theorem 12.11.2 (Gauss–Bonnet). The Euler characteristic χ(M)of a compact surface M satisfies

∫

M

Kp dAM = 2π χ(M), (12.11.1)

where Kp is the Gaussian curvature at the point p ∈M .

Here we no longer assume that the curvature is positive.

Remark 12.11.3. The relation (12.11.1) is similar to the line inte-gral expression for the rotation index in formula (13.1.1).

One way of proving Gauss–Bonnet for embedded surfaces is to usethe notion of algebraic degree for maps between surfaces, similar to thealgebraic degree of a self-map of a circle. We note the following.

(1) The Gauss–Bonnet theorem holds for all surfaces, whether ori-entable or not.

(2) For the real projective plane RP2 we have χ(RP2) = 1. There-fore for every metric on RP2 we have

∫RP2 K dARP2 = 2π.

(3) For the torus T 2 we have χ(T 2) = 0. Therefore for every metricon T 2 we have

∫T 2 K dAT 2 = 0. It follows that the torus does

not admit any metric of positive Gaussian curvature.

12.11.1. Miscellaneous. This material is optional. For a reader fa-miliar with elements of Riemannian geometry including Jacobi fields, it isworth mentioning that the Jacobi equation y′′ +Ky = 0 of a Jacobi field yon M (expressing an infinitesimal variation by geodesics) sheds light on thenature of curvature in a way that no mere formula for K could. Thus,in positive curvature, geodesics converge, while in negative curvature, theydiverge. However, to prove the Jacobi equation, one would need to havealready an intrinsically well-defined quantity on the left hand side, y′′+Ky,of the Jacobi equation. In particular, one would need an already intrinsicnotion of curvature K. Thus, a proof of the theorema egregium necessarilyprecedes the deeper insights provided by the Jacobi equation. Similarly, theGaussian curvature at p ∈ M is the first significant term in the asymptoticexpansion of the length of a “small” circle of center p. This fact, too, shedsmuch light on the nature of Gaussian curvature. However, to define whatone means by a “small” circle, requires introducing higher order notions suchas the exponential map, which are usually understood at a later stage thanthe notion of Gaussian curvature, cf. [Ca76, Car92, Ch93, GaHL04].


12.11.2. Binet–Cauchy identity. This material is optional.

Theorem 12.11.4 (Binet–Cauchy identity). The 3-dimensional case ofthe Binet–Cauchy identity is the identity (a · c)(b · d) = (a · d)(b · c) + (a ×b) · (c× d), where a, b, c, and d are vectors in R3.

The formula can also be written as a formula giving the dot product oftwo wedge products, namely (a× b) · (c× d) = (a · c)(b · d)− (a · d)(b · c).

A special case of Binet–Cauchy is the case of vectors a = c and b = d,when we obtain |a× b|2 = |a|2|b|2 − |a · b|2.

When both vectors a, b are unit vectors, we obtain the usual relation1 = cos2(φ) + sin2(φ) (here the vector product gives the sine and the scalarproduct gives the cosine) where φ is the angle between the vectors.

Let a = x1 and b = x2 be the two tangent vectors to the surface withparametrisation x(u1, u2). Then |x1 × x2|2 = g11g22 − g212 = det(gij).

Equivalently, we have |x1 × x2| =√det(gij). This can be thought of as

the area of the parallelogram spanned by the two vectors.

CHAPTER 13

Rotation index, isothermisation, pseudosphere

13.1. Rotation index of closed curves and total curvature

The rotation index was defined in Section 11.6 where we mostlydealt with Jordan curves. For general regular closed curves (with pos-sible self-intersections) we have the following result.

The result is analogous to the relation between the Euler character-istic of a surface and its total Gaussian curvature (see Section 12.11).

Theorem 13.1.1. Let α(s) be an arclength parametrisation of ageometric curve C ⊆ R2. Then the rotation index ια is related to the

total signed curvature Tot(C) as follows:

Tot(C) = 2πια. (13.1.1)

Proof. The proof is similar to that given in Section 11.6. Weexploit the continuous branch θ(s) = 1

ilogα′(s), s ∈ [0, L] as in The-

orem 11.5.4. Applying the change of variable formula for integration,we obtain

Tot(C) =

∮

C

kα(s)ds =

∮

C

dθ

dsds =

∫ 2πια

0

dθ = 2πια,

where the upper limit of integration is 2πια (by definition of the rotationindex). �

Example 13.1.2. The rotation index of a figure-8 curve (known asthe lemniscate of Gerono, one of the Lissajous curves) is 0. Hence itstotal signed curvature vanishes.

Example 13.1.3. Describe an immersed curve with rotation in-dex 2.

13.2. Surfaces of revolution and isothermalisation

Below we will express the metric of a surface of revolution in isother-mal coordinates. Recall that if a surface is obtained by revolving acurve (r(φ), z(φ)), we obtain metric coefficients g11 = r2 and g22 =(drdφ

)2+(dzdφ

)2. Thus we obtain the following theorem.

157

158 13. ROTATION INDEX, ISOTHERMISATION, PSEUDOSPHERE

Theorem 13.2.1. We have the following expression of the metricof a surface of revolution in coordinates u1 = r, u2 = φ:

g11 = r2, g22 =(drdφ

)2+(dzdφ

)2, g12 = 0. (13.2.1)

We will use this theorem to carry out explicit isothermalisation inthe case of a surface of revolution.

Proposition 13.2.2. Let (r(φ), z(φ)), where r(φ) > 0, be an ar-clength parametrisation of the generating curve of a surface of revolu-tion M . Then the change of variable

ψ =

∫1

r(φ)dφ,

produces an isothermal parametrisation ofM in terms of variables (θ, ψ).With respect to the new coordinates, the first fundamental form is givenby a scalar matrix with metric coefficients r(φ(ψ))2δij.

Remark 13.2.3. The existence of such a parametrisation is pre-dicted by the uniformisation theorem (Theorem 13.12.2) in the case ofa general surface but the general result is less explicit.

Proof of Proposition 13.2.2. We will carry out a change ofvariables affecting only the second variable φ.

Step 1. Consider an arbitrary monotone change of parameter φ =φ(ψ). By chain rule,

dr

dψ=dr

dφ

dφ

dψ.

Step 2. Consider the (possibly non-scalar) first fundamental formas given by (13.2.1). We need to impose the condition

g11 = g22 (13.2.2)

to ensure that the matrix of metric coefficients be a scalar matrix. ByTheorem 13.2.1, equality (13.2.2) is equivalent to the equation

r2 =

(dr

dψ

)2

+

(dz

dψ

)2

. (13.2.3)

Step 3. By chain rule, (13.2.3) is equivalent to the formula r2 =((drdφ

)2+(dzdφ

)2)(dφdψ

)2, or

r =

√(dr

dφ

)2

+

(dz

dφ

)2dφ

dψ. (13.2.4)

13.3. GAUSSIAN CURVATURE OF PSEUDOSPHERE 159

Step 4. In the case when the original generating curve is para-metrized by arclength (as in our proposition), equation (13.2.4) be-comes r = dφ

dψ. Solving for ψ, we obtain ψ(φ) =

∫dφr(φ)

.

Step 5. Since the change of parameter is monotone, we can solvethe equation ψ = ψ(φ) for φ obtaining φ = φ(ψ). Substituting thenew variable φ(ψ) in place of φ in the parametrisation of the surface,we obtain a new parametrisation of the surface of revolution in coordi-nates (θ, ψ).

Step 6. With respect to the new parametrisation, the first fun-damental form is scalar with conformal factor λ = r2(φ(ψ)) as perequation (13.2.3). �

13.3. Gaussian Curvature of pseudosphere

As an application of the isothermalisation of Section 13.2, we cal-culate the curvature of the pseudosphere. The pseudosphere (see Sec-tion 5.10) is the surface of revolution generated by functions r(φ) = eφ

and z(φ) =∫ φ0

√1− e2τdτ = −

∫ 0

φ

√1− e2τdτ , where −∞ < φ ≤ 0.

Its metric coefficients are given by the matrix (gij) =

(e2φ 00 1

), which

is not a scalar matrix.

Theorem 13.3.1. The pseudosphere has constant Gaussian curva-ture K = −1.

Proof. Instead of applying the general formula for curvature, weuse the trick of a change of coordinates that results in isothermal co-ordinates so we can apply the formula for curvature in terms of theLaplace–Beltrami operator. We apply Proposition 13.2.2 to introducethe change of coordinates

ψ =

∫dφ

r(φ)=

∫e−φdφ = −e−φ. (13.3.1)

From (13.3.1) we obtain φ = − ln(−ψ). This results in isothermalcoordinates (θ, ψ) with conformal factor f(ψ) = r(φ(ψ)). Therefore

f(ψ) = r(φ(ψ)) = e− ln(−ψ) =1

eln(−ψ)=

1

−ψ = (−ψ)−1.

We have λ = f 2 = 1ψ2 . Recall that the metric with gij(θ, ψ) =

1ψ2 δij is

by definition hyperbolic (here we merely use the variables θ, ψ in placeof x, y). Therefore it has curvature K = −1 as shown in Section 12.3.For completeness we carry out the calculation as follows. In isother-mal coordinates (θ, ψ), Gaussian curvature is given by the formula in


terms of the Laplace–Beltrami operator. The Gaussian curvature ofthe pseudosphere is

K = −∆LB ln f

= −1

λ

∂2

∂ψ2ln f(ψ)

= −ψ2 ∂2

∂ψ2ln((−ψ)−1)

= ψ2 ∂2

∂ψ2ln(−ψ).

Differentiating with respect to ψ, we obtain

K = ψ2 ∂

∂ψ

(−1

−ψ

)

= ψ2 ∂

∂ψ

(1

ψ

)

= ψ2

(− 1

ψ2

)

= −1,

as required. �

13.4. Transition from classical to modern diff geom

Remark 13.4.1. The theorema egregium of Gauss marks the tran-sition from classical differential geometry of curves and surfaces em-bedded in 3-space, to modern differential geometry of surfaces (andmanifolds) studied intrinsically.

More specifically, once we have a notion of Gaussian curvature (andmore generally sectional curvature) that only depends on the metricon a surface (or manifold), we can study the geometry of the surface(or manifold) intrinsically, i.e., without any reference to a Euclideanembedding.

Remark 13.4.2. To formulate the intrinsic viewpoint, one needsthe notion of duality of vector and covector. This theme is treated inSection 13.5.

13.5. Duality in linear algebra; 1-forms

Let V be a finite-dimensional real vector space.

Example 13.5.1. Euclidean space Rn is an example of a real vectorspace of dimension n with basis (e1, . . . , en).

13.6. DUAL VECTOR SPACE 161

Example 13.5.2. The tangent plane TpM of a regular surface Mat a point p ∈M is a real vector space of dimension 2.

Definition 13.5.3. A linear form, also called 1-form, φ on V is alinear functional

φ : V → R.

Definition 13.5.4 (dx, dy). In the usual Euclidean plane of vectors

v = v1e1 + v2e2

represented by arrows, we denote by dx the 1-form which extracts theabscissa of the vector, and by dy the 1-form which extracts the ordinateof the vector:

dx(v) = v1,

anddy(v) = v2.

Example 13.5.5. For a vector v = 3e1+4e2 with components (3, 4)we obtain

dx(v) = 3, dy(v) = 4.

Definition 13.5.6. The quadratic forms dx2 and dy2 are definedby squaring the value of the 1-form on v:

dx2(v) = (dx(v))2.

Such quadratic forms are called rank-1 quadratic forms.

Thus,dx2(v) = (v1)2, dy2(v) = (v2)2.

Example 13.5.7. In the case v = 3e1 + 4e2 we obtain dx2(v) = 9and dy2(v) = 16.

13.6. Dual vector space

Definition 13.6.1. The dual space of V , denoted V ∗, is the spaceof all linear forms on V :

V ∗ = {φ : φ is a 1-form on V } .Evaluating φ at an element x ∈ V produces a scalar φ(x) ∈ R.

Definition 13.6.2. The evaluation map1 is the natural pairing be-tween V and V ∗, namely a linear map denoted

〈 , 〉 : V × V ∗ → R,

defined by evaluating y at x, i.e., setting 〈x, y〉 = y(x), for all x ∈ Vand y ∈ V ∗.

1hatzava?


Remark 13.6.3. Note we are using the same notation for the pair-ing as for the scalar product in Euclidean space. The notation is quitewidespread.

Definition 13.6.4. If V admits a basis of vectors

(xi) = (x1, x2, . . . , xn),

then the dual space V ∗ admits a unique basis, called the dual ba-sis (yj) = (y1, . . . , yn), satisfying

〈xi, yj〉 = δij , (13.6.1)

for all i, j = 1, . . . , n, where δij is the Kronecker delta function.

Example 13.6.5. Let V = R2. We have the standard basis e1, e2for V . The 1-forms dx, dy form a basis for the dual space V ∗. Thenthe basis (dx, dy) is the dual basis to the basis (e1, e2).

Example 13.6.6. Let V = R2 identified with C for convenience.We have the basis (1, e

iπ3 ) for V where 1 = e1 + 0e2 while

eiπ3 =

1

2e1 +

√3

2e2.

Find the dual basis in V ∗. The answer is y1 = dx− 1√3dy, y2 =

2√3dy.

13.7. Derivations

Let E be a space of dimension n, and let p ∈ E be a fixed point.The following definition is independent of coordinates.

Definition 13.7.1. Let

Dp = {f : f ∈ C∞}be the ring of real C∞-functions f defined in an arbitrarily small openneighborhood of p ∈ E.

Remark 13.7.2. The ring operations are pointwise multiplicationand pointwise addition, where we choose the intersection of the twodomains as the domain of the new function.

Note that Dp is infinite-dimensional as it includes all polynomials.

Theorem 13.7.3. Choose coordinates (u1, . . . , un) in E. A partialderivative ∂

∂uiat p is a 1-form

∂

∂ui: Dp → R

13.8. CHARACTERISATION OF DERIVATIONS 163

on the space Dp, satisfying Leibniz rule

∂(fg)

∂ui

∣∣∣∣p

=∂f

∂ui

∣∣∣∣p

g(p) + f(p)∂g

∂ui

∣∣∣∣p

(13.7.1)

for all f, g ∈ Dp.

Formula (13.7.1) can be written briefly as

∂∂ui

(fg) = ∂∂ui

(f)g + f ∂∂ui

(g),

with the understanding that both sides are evaluated at the point p.This was proved in calculus. Formula (13.7.1) motivates the followingmore general definition of a derivation.

Definition 13.7.4. A derivation X at the point p ∈ E is a 1-form

X : Dp → R

on the space Dp satisfying Leibniz rule:

X(fg) = X(f)g(p) + f(p)X(g) (13.7.2)

for all f, g ∈ Dp.

Remark 13.7.5. Linearity of a derivation is required only withregard to scalars in R, not with respect to functions.

13.8. Characterisation of derivations

It turns out that the space of derivations is spanned by partialderivatives.

Proposition 13.8.1. Let E be an n-dimensional space, and p ∈E. Then the collection of all derivations at p is a vector space ofdimension n.

Proof in case n = 1. We will first prove the result in the case n =1 of a single variable u at the point p = 0. Let X : Dp → R be aderivation. Then X(1) = X(1 · 1) = 2X(1) by Leibniz rule. There-fore X(1) = 0, and similarly for any constant by linearity of X.

Now consider the monic polynomial u = u1 of degree 1, viewedas a linear function u ∈ Dp=0. We evaluate the derivation X at theelement u ∈ D and set c = X(u). By the Taylor remainder formula, anyfunction f ∈ Dp=0 can be written as f(u) = a+ bu+ g(u)u where b =


∂f∂u(0) and g is smooth and g(0) = 0. Now we have by linearity

X(f) = X(a+ bu+ g(u)u)

= bX(u) +X(g)u(0) + g(0) ·X(u)

= bc+ 0 + 0

= c∂

∂u(f).

Thus the derivation X coincides with the derivation c ∂∂u

for all f ∈Dp. Hence the tangent space is 1-dimensional and spanned by theelement ∂

∂u, proving the theorem in this case. �

2

13.9. Tangent space and cotangent space

Definition 13.9.1. Let p ∈ E. The space of derivations at p iscalled the tangent space Tp = TpE at p.

The results of the previous section can be formulated as follows.

2Proof in case n = 2. This material is optional. Let us prove theresult in the case n = 2 of two variables u, v at the origin p = (0, 0). Let Xbe a derivation. Then X(1) = X(1 · 1) = 2X(1) by Leibniz rule. There-foreX(1) = 0, and similarly for any constant by linearity ofX. Now considerthe monic polynomial u = u1 of degree 1, i.e., the linear function u ∈ Dp=0.We evaluate the derivation X at u and set c = X(u). Similarly, considerthe monic polynomial v = v1 of degree 1, i.e., the linear function v ∈ Dp=0.We evaluate the derivation X at v and set c = X(v). By the Taylor re-mainder formula, any function f ∈ Dp=0, where f = f(u, v), can be written

as f(u, v) = a+ bu+ bv + g(u, v)u2 + h(u, v)uv + k(u, v)v2, where the func-tions g(u, v), h(u, v), and k(u, v) are smooth. Note that the coefficients b

and b are the first partial derivatives of f at the origin. Now we have

X(f) = X(a+ bu+ bv + g(u, v)u2 + h(u, v)uv + k(u, v)v2)

= bX(u) + bX(v) +(X(g)u2 + g(u, v)2uX(u) + · · ·

)

= bX(u) + bX(v)

= bc+ bc

= c∂

∂u(f) + c

∂

∂v(f)

by evaluating at the point (0, 0). Thus the derivation X coincides with thederivation c ∂∂u+c

∂∂v for all test functions f ∈ Dp. Hence the two partials span

the tangent space. Therefore the tangent space is 2-dimensional, proving thetheorem in this case.

13.10. CONSTRUCTING BILINEAR FORMS OUT OF 1-FORMS 165

Corollary 13.9.2. Let (u1, . . . , un) be coordinates for E, and let p ∈E. Then a basis for the tangent space Tp is given by the n partialderivatives (

∂

∂ui

), i = 1, . . . , n.

Definition 13.9.3. The space dual to the tangent space Tp is calledthe cotangent space, and denoted T ∗

p .

Definition 13.9.4. An element of a tangent space is a vector, whilean element of a cotangent space is called a 1-form, or a covector.

Definition 13.9.5. The basis dual to the basis(∂∂ui

)is denoted

(duj), j = 1, . . . , n.

Thus each duj is by definition a linear form, or cotangent vector(covector for short). We are therefore working with dual bases

(∂∂ui

)

for vectors, and (duj) for covectors. The evaluation map as in (13.6.1)gives ⟨

∂

∂ui, duj

⟩= duj

(∂

∂ui

)= δji , (13.9.1)

where δji is the Kronecker delta.

13.10. Constructing bilinear forms out of 1-forms

Recall that the polarisation formula (see Definition 1.4.3) allowsone to reconstruct a symmetric bilinear form B = B(v, w), from thequadratic form Q(v) = B(v, v), at least if the characteristic is not 2:

B(v, w) =1

4(Q(v + w)−Q(v − w)). (13.10.1)

Similarly, one can construct bilinear forms out of the 1-forms dui,as follows.

Example 13.10.1. Consider a quadratic form ai(dui)2 defined by

a linear combination of the rank-1 quadratic forms (dui)2, as in Defi-nition 13.5.6.

Polarizing the quadratic form, one obtains a bilinear form on thetangent space Tp.

Example 13.10.2. Let v = v1e1 + v2e2 be an arbitrary vector inthe plane. Let dx and dy be the standard covectors, extracting, respec-tively, the first and second coordinates of v. Consider the quadratic


form Q given by Q = Edx2 + Fdy2, where E,F ∈ R. Here Q(v) iscalculated as

Q(v) = E(dx(v))2 + F (dy(v))2 = E(v1)2 + F (v2)2.

Polarisation then produces the bilinear form B = B(v, w), where vand w are arbitrary vectors, given by the formula

B(v, w) = E dx(v) dx(w) + F dy(v) dy(w).

Example 13.10.3. Setting E = F = 1 in the previous example, weobtain the standard scalar product in the plane:

B(v, w) = v · w = dx(v) dx(w) + dy(v) dy(w) = v1w1 + v2w2.

13.11. First fundamental form

Definition 13.11.1. A metric (or first fundamental form) g is asymmetric bilinear form on the tangent space at p, namely g : Tp×Tp →R, defined for all p and varying continuously and smoothly in p.

Remark 13.11.2. In Riemannian geometry one requires the asso-ciated quadratic form to be positive definite. In relativity theory oneuses a form of type (3, 1).

Recall that the basis for Tp in coordinates (ui) is given by thetangent vectors ∂

∂ui. These are given by certain derivations (see Sec-

tion 13.7).The first fundamental form g is traditionally expressed by a matrix

of coefficients called metric coefficients gij, giving the inner product ofthe i-th and the j-th vector in the basis: gij = g

(∂∂ui, ∂∂uj

), where g is

the first fundamental form.

Definition 13.11.3. The square norm the i-th vector is given by

the coefficient gii =∥∥ ∂∂ui

∥∥2.We will express the first fundamental form in more intrinsic notation

of quadratic forms built from 1-forms (covectors).

13.12. Dual bases in differential geometry

Let us now restrict attention to the case of 2 dimensions, i.e., thecase of surfaces. At every point p = (u1, u2), we have the metric coef-ficients gij = gij(u

1, u2). Each metric coefficient is thus a function oftwo variables.

We will only consider the case when the matrix is diagonal. This canalways be achieved, in two dimensions, at a point by a suitable changeof coordinates, by the uniformisation theorem (see Theorem 13.12.2).

13.13. MORE ON DUAL BASES 167

We set x = u1 and y = u2 to simplify notation. In the notationdeveloped in Section 13.10, we can write the quadratic form associatedwith the first fundamental form as follows:

g = g11(x, y)(dx)2 + g22(x, y)(dy)

2. (13.12.1)

For example, if the metric coefficients form an identity matrix: gij =δij, we obtain the standard flat metric

g = (dx)2 + (dy)2 (13.12.2)

or simply g = dx2 + dy2.

Example 13.12.1 (Hyperbolic metric). Let g11 = g22 =1y2

at each

point (x, y) where y > 0. This means that∣∣ ∂∂u1

∣∣ = 1yand

∣∣ ∂∂u2

∣∣ = 1y.

The resulting hyperbolic metric in the upper half plane {y > 0} is ex-pressed by the quadratic form 1

y2(dx2 + dy2). Note that this expression

is undefined whenever y = 0. See Section 12.3. The hyperbolic metricin the upper half plane is a complete metric.

Closely related results are the Riemann mapping theorem and theconformal representation theorem.

Theorem 13.12.2 (Riemann mapping/uniformisation). Every met-ric on a connected3 surface is conformally equivalent to a metric ofconstant Gaussian curvature.

From the complex analytic viewpoint, the uniformisation theoremstates that every Riemann surface is covered by either the sphere, theplane, or the upper halfplane. Thus no notion of curvature is needed forthe statement of the uniformisation theorem. However, from the dif-ferential geometric point of view, what is relevant is that every confor-mal class of metrics contains a metric of constant Gaussian curvature.See [Ab81] for a lively account of the history of the uniformisationtheorem. A lot of information on the uniformisation theorem and theRiemann mapping theorem can be found at https://mathoverflow.net/q/10516.

13.13. More on dual bases

Recall that if (x1, . . . , xn) is a basis for a vector space V then thedual vector space V ∗ possesses a basis called the dual basis and de-noted (y1, . . . , yn) satisfying 〈xi, yj〉 = yj(xi) = δij .

3kashir

https://mathoverflow.net/q/10516

https://mathoverflow.net/q/10516


Example 13.13.1. In R2 we have a basis (x1, x2) =(∂∂x, ∂∂y

)in the

tangent plane Tp at a point p. The dual basis of 1-forms (y1, y2) for T∗p

is denoted (dx, dy). Thus we have⟨∂

∂x, dx

⟩= dx

(∂

∂x

)= 1

and ⟨∂

∂y, dy

⟩= dy

(∂

∂y

)= 1,

while ⟨∂

∂x, dy

⟩= dy

(∂

∂x

)= 0,

etcetera.

Similarly, in polar coordinates at a point p 6= 0 we have a basis(∂∂r, ∂∂θ

)for Tp, and a dual basis (dr, dθ) for T ∗

p . Thus we have⟨∂

∂r, dr

⟩= dr

(∂

∂r

)= 1

and ⟨∂

∂θ, dθ

⟩= dθ

(∂

∂θ

)= 1,

while ⟨∂

∂r, dθ

⟩= dθ

(∂

∂r

)= 0,

etcetera.Now in polar coordinates we have a natural area element r dr dθ.

Area of a region D is calculated by Fubini’s theorem as∫∫

D

r dr dθ =

∫ (∫rdr

)dθ.

Thus we have a natural basis (y1, y2) = (rdr, dθ) in T ∗p when p 6= 0,

i.e., y1 = rdr while y2 = dθ. Its dual basis (x1, x2) in Tp can be easilyidentified. It is

(x1, x2) =

(1

r

∂

∂r,∂

∂θ

).

Indeed, we have

〈x1, y1〉 =⟨1

r

∂

∂r, rdr

⟩= rdr

(1

r

∂

∂r

)= r

1

rdr

(∂

∂r

)= 1,

etcetera.

CHAPTER 14

Lattices and tori

14.1. Circle via the exponential map

We will discuss lattices in Euclidean space Rb in Section 14.2. Herewe give an intuitive introduction in the simplest case b = 1. Everylattice (discrete1 subgroup; see definition below in Section 14.2) in R =R1 is of the form

Lα = αZ = {nα : n ∈ Z} ⊆ R,

for some real α > 0. It is spanned by the vector αe1 (or −αe1).The lattice Lα ⊆ R is in fact an additive subgroup. Therefore

we can form the quotient group R/Lα. This quotient is a circle (seeTheorem 14.1.1). The 1-volume, i.e. the length, of the quotient circleis precisely α. We will give a description in terms of the complexfunction ez.

Theorem 14.1.1. The quotient group R/Lα is isomorphic to thecircle S1 ⊆ C.

Proof. Consider the map φ : R → C defined by

φ(x) = ei2πxα .

By the usual addition rule for the exponential function, this map is ahomomorphism from the additive structure on R to the multiplicativestructure in the group C \ {0}. Namely, we have

(∀x ∈ R)(∀y ∈ R) φ(x+ y) = φ(x)φ(y).

Furthermore, we have φ(x + αm) = φ(x) for all m ∈ Z and there-

fore ker φ = Lα. By the group-theoretic isomorphism theorem, themap φ descends to a map

φ : R/Lα → C,

which is injective. Its image is the unit circle S1 ⊆ C, which is a groupunder multiplication. �

1b’didah

169

170 14. LATTICES AND TORI

14.2. Lattice, fundamental domain

Let b > 0 be an integer.

Definition 14.2.1. A lattice L ⊆ Rb is the integer span of a linearlyindependent set of b vectors.

Thus, if vectors v1, . . . , vb are linearly independent, then they spana lattice

L = {n1v1 + . . .+ nbvb : ni ∈ Z} = Zv1 + Zv2 · · ·+ Zvb

Note that the subgroup is isomorphic to Zb.

Definition 14.2.2. An orbit of a point x0 ∈ Rb under the actionof a lattice L is the subset of Rb given by the collection of elements

{x0 + g : g ∈ L} .These can also be viewed as the cosets of the lattice in Rb.

Definition 14.2.3. The quotient

Rb/L

is called a b-torus.

At this point tori are understood at the group-theoretic level as inthe case of the circle R/L.

14.3. Fundamental domain

Definition 14.3.1. A fundamental domain for the torus Rb/L is aclosed set F ⊆ Rb satisfying the following three conditions:

• every orbit meets F in at least one point;• every orbit meets the interior Int(F ) of F in at most one point;• the boundary ∂F is of zero b-dimensional volume (and can bethought of as a union of (b− 1)-dimensional hyperplanes).

In the literature, one often replaces “n-dimensional volume” by n-dimensional “Lebesgue measure”.

Example 14.3.2. The interval [0, α] is a fundamental domain forthe circle R/Lα.

Example 14.3.3. The parallelepiped spanned by a collection ofbasis vectors for L ⊆ Rb is a fundamental domain fot L.

More concretely, consider the following example.

14.4. LATTICES IN THE PLANE 171

Example 14.3.4. The vectors e1 and e2 in R2 span the unit squarewhich is a fundamental domain for the lattice Z2 ⊆ R2 = C of Gaussianintegers.

Example 14.3.5. Consider the vectors v = (1, 0) and w = (12,√32)

in R2 = C. Their span is a parallelogram giving a fundamental domainfor the lattice of Eisenstein integers (see Example 14.4.2).

Definition 14.3.6. The total volume of the b-torus Rb/L is bydefinition the b-volume of a fundamental domain.

It is shown in advanced calculus that the total volume thus definedis independent of the choice of a fundamental domain.

14.4. Lattices in the plane

Let b = 2. Every lattice L ⊆ R2 is of the form

L = SpanZ(v, w) ⊆ R2,

where {v, w} is a linearly independent set. For example, let α and βbe nonzero reals. Set

Lα,β = SpanZ(αe1, βe2) ⊆ R2.

This lattice admits an orthogonal basis, namely {αe1, βe2}.

Example 14.4.1 (Gaussian integers). For the standard lattice Zb ⊆Rb, the torus Tb = Rb/Zb satisfies vol(Tb) = 1 as it has the unit cubeas a fundamental domain.

In dimension 2, the resulting lattice in C = R2 is called theGaussianintegers LG. It contains 4 elements of least length. These are the fourthroots of unity. We have

LG = SpanZ(1, i) ⊆ C.

Example 14.4.2 (Eisenstein integers). Consider the lattice LE ⊆R2 = C spanned by 1 ∈ C and the sixth root of unity e

2πi6 ∈ C:

LE = SpanZ(eiπ/3, 1) = Z eiπ/3 + Z 1 ⊆ C. (14.4.1)

The resulting lattice is called the Eisenstein integers. The torus T2 =

R2/LE satisfies area(T2) =√32. The Eisenstein lattice contains 6 ele-

ments of least length, namely all the sixth roots of unity.


14.5. Successive minima of a lattice

Let B be Euclidean space, and let ‖ ‖ be the Euclidean norm.Let L ⊆ (B, ‖ ‖) be a lattice, i.e., span of a collection of b linearlyindependent vectors where b = dim(B).

Definition 14.5.1. The first successive minimum, λ1(L, ‖ ‖) is theleast length of a nonzero vector in L.

We can express the definition symbolically by means of the formula

λ1(L, ‖ ‖) = min{‖v1‖

∣∣ v1 ∈ L \ {0}}.

We illustrate the geometric meaning of λ1 in terms of the circle ofTheorem 14.1.1.

Theorem 14.5.2. Consider a lattice L ⊆ R. Then the circle R/Lsatisfies

length(R/L) = λ1(L).

Proof. This follows by choosing the fundamental domain F =[0, α] where α = λ1(L), so that L = αZ, cf. Example 14.2 above. �

Remark 14.5.3. When α = 1, we can choose a representative fromthe orbit of x to be the fractional part {x} of x.

Definition 14.5.4. For k = 2, define the second successive min-imum of the lattice L with rank(L) ≥ 2 as follows. Given a pair ofvectors S = {v, w} in L, define the size2 |S| of S by setting

|S| = max(‖v‖, ‖w‖).Then the second successive minimum, λ2(L, ‖ ‖) is the least size of apair of non-proportional vectors in L:

λ2(L) = infS|S|,

where S runs over all linearly independent (i.e. non-proportional) pairsof vectors {v, w} ⊆ L.

Example 14.5.5. For both the Gaussian and the Eisenstein integerswe have λ1 = λ2 = 1.

Example 14.5.6. For the lattice Lα,β we have λ1(Lα,β) = min(|α|, |β|)and λ2(Lα,β) = max(|α|, |β|).

2Quotation marks: merka’ot.

14.7. SPHERE AND TORUS AS TOPOLOGICAL SURFACES 173

14.6. Gram matrix

The volume of the torus Rb/L (see Definition 14.2.3) is also calledthe covolume of the lattice L. It is by definition the volume of a funda-mental domain for L, e.g. a parallelepiped spanned by a Z-basis for L.

The Gram matrix was defined in Definition 5.6.2.

Theorem 14.6.1. Let L ⊂ Rb be a lattice spanned by linearly inde-pendent vectors (v1, . . . , vb). Then the volume of the torus Rb/L is thesquare root of the determinant of the Gram matrix Gram(v1, . . . , vb).

In geometric terms, the parallelepiped P spanned by the vectors {vi}satisfies

vol(P ) =√det(Gram(S)). (14.6.1)

Proof. Let A be the square matrix whose columns are the columnvectors v1, v2 . . . , vn in Rn. It is shown in linear algebra that

vol(P ) = |det(A)|.Let B = AtA, and let B = (bij). Then

bij = vti vj = 〈vi, vj〉Hence B = Gram(S). Thus

det(Gram(S)) = det(At A) = det(A)2 = vol(P )2


14.7. Sphere and torus as topological surfaces

The topology of surfaces will be discussed in more detail in Chap-ter 17. For now, we will recall that a compact surface can be eitherorientable or non-orientable. An orientable surface is characterizedtopologically by its genus, i.e. number of “handles”.

Recall that the unit sphere in R3 can be represented implicitly bythe equation

x2 + y2 + z2 = 1.

Parametric representations of surfaces are discussed in Section 5.3.

Example 14.7.1. The sphere has genus 0 (no handles).

Theorem 14.7.2. The 2-torus is characterized topologically in oneof the following four equivalent ways:

(1) the Cartesian product of a pair of circles: S1 × S1;(2) the surface of revolution in R3 obtained by starting with the

following circle in the (x, z)-plane: (x − 10)2 + z2 = 1 (forexample), and rotating it around the z-axis;


(3) a quotient R2/L of the plane by a lattice L;(4) a compact 2-dimensional manifold of genus 1.

The equivalence between items (2) and (3) can be seen by markinga pair of generators of L by different color, and using the same colors toindicate the corresponding circles on the embedded torus of revolution,as follows:

Figure 14.7.1. Torus viewed by means of its lattice(left) and by means of a Euclidean embedding (right)

Note by comparison that a circle can be represented either by itsfundamental domain which is [0, 2π] (with endpoints identified), or asthe unit circle embedded in the plane.3

14.8. Standard fundamental domain

We will discuss the case b = 2 in detail. An important role is playedin this dimension by the standard fundamental domain.

3The Hermite constant γb is defined in one of the following two equivalent ways:

(1) γb is the square of the maximal first successive minimum λ1, among alllattices of unit covolume;

(2) γb is defined by the formula

√γb = sup

{λ1(L)

vol(Rb/L)1

b

∣∣∣∣L ⊆ (Rb, ‖ ‖)}, (14.7.1)

where the supremum is extended over all lattices L in Rb with a Euclideannorm ‖ ‖.

A lattice realizing the supremum may be thought of as the one realizing the densestpacking in Rb when we place the balls of radius 1

2λ1(L) at the points of L. Indimensions b ≥ 3, the Hermite constants are harder to compute, but explicit values(as well as the associated critical lattices) are known for small dimensions, e.g. γ3 =

21

3 = 1.2599..., while γ4 =√2 = 1.4142....

14.9. CONFORMAL PARAMETER τ OF A LATTICE 175

Definition 14.8.1. The standard fundamental domain, denotedD,is the set

D =

{z ∈ C

∣∣∣∣ |z| ≥ 1, |Re(z)| ≤ 1

2, Im(z) > 0

}(14.8.1)

cf. [Ser73, p. 78].

The domain D a fundamental domain for the action of PSL(2,Z)in the upper half-plane of C.

Lemma 14.8.2. Multiplying a lattice L ⊆ C by nonzero complexnumbers does not change the value of the quotient

λ1(L)2

area(C/L).

Proof. We write such a complex number as reiθ. Note that multi-plication by reiθ can be thought of as a composition of a scaling by thereal factor r, and rotation by angle θ. The rotation is an isometry (con-gruence) that preserves all lengths, and in particular the length λ1(L)and the area of the quotient torus.

Meanwhile, multiplication by r results in a cancellation

λ1(rL)2

area(C/rL)=

(rλ1(L))2

r2 area(C/L)=

r2λ1(L)2

r2 area(C/L)=

λ1(L)2

area(C/L),


14.9. Conformal parameter τ of a lattice

Two lattices in C are said to be similar if one is obtained from theother by multiplication by a nonzero complex number.

Theorem 14.9.1. Every lattice in C is similar to a lattice spannedby {τ, 1} where τ is in the standard fundamental domain D of (14.8.1).The value τ = eiπ/3 corresponds to the Eisenstein integers (14.4.1).

Proof. Let L ⊆ C be a lattice. Choose a “shortest” vector z ∈L, i.e. we have |z| = λ1(L). By Lemma 14.8.2, we may replace thelattice L by the lattice z−1L.

Thus, we may assume without loss of generality that the com-plex number +1 ∈ C is a shortest element in the lattice L. Thuswe have λ1(L) = 1. Now complete the element +1 to a Z-basis

{τ ,+1}for L. Here we may assume, by replacing τ by−τ if necessary, that Im(τ) >0.


Now consider the real part Re(τ). We adjust the basis by adding asuitable integer k to τ :

τ = τ − k where k =[Re(τ) + 1

2

](14.9.1)

(the brackets denote the integer part), so it satisfies the condition

−1

2≤ Re(τ) ≤ 1

2.

Since τ ∈ L, we have |τ | ≥ λ1(L) = 1. Therefore the element τ lies inthe standard fundamental domain (14.8.1). �

Example 14.9.2. For the “rectangular” lattice Lα,β = SpanZ(α, βi),we obtain

τ(Lα,β) =

{ |β||α| i if |β| > |α||α||β| i if |α| > |β|.

Corollary 14.9.3. Let b = 2. Then we have the following valuefor the Hermite constant: γ2 = 2√

3= 1.1547.... The corresponding

optimal lattice is homothetic to the Z-span of cube roots of unity in C(i.e. the Eisenstein integers).

Proof. Choose τ as in (14.9.1) above. The pair

{τ,+1}

is a basis for the lattice. The imaginary part satisfies Im(τ) ≥√32, with

equality possible precisely for

τ = eiπ3 or τ = ei

2π3 .

Moreover, if τ = r exp(iθ), then

|τ | sin θ = Im(τ) ≥√3

2.

The proof is concluded by calculating the area of the parallelogramin C spanned by τ and +1;

λ1(L)2

area(C/L)=

1

|τ | sin θ ≤ 2√3,


Definition 14.9.4. A τ ∈ D is said to be the conformal parameterof a flat torus T 2 if T 2 is similar to a torus C/L where L = Zτ + Z1.

14.10. CONFORMAL PARAMETER τ OF TORI OF REVOLUTION 177

14.10. Conformal parameter τ of tori of revolution

The results of Section 13.2 have the following immediate conse-quence.

Corollary 14.10.1. Consider a torus of revolution in R3 formedby rotating a Jordan curve of length L > 0, with unit speed param-etisation (f(φ), g(φ)) where φ ∈ [0, L]. Then the torus is conformallyequivalent to a flat torus

R2/Lc,d.

Here R2 is the (θ, ψ)-plane, where ψ is the antiderivative of 1f(φ)

as in

Section 13.2; while the rectangular lattice Lc,d ⊂ R2 is spanned by theorthogonal vectors c ∂

∂θand d ∂

∂ψ, so that

Lc,d = Span

(c∂

∂θ, d

∂

∂ψ

)= cZ⊕ dZ,

where c = 2π and d =∫ L0

dφf(φ)

.

Figure 14.10.1. Torus: lattice (left) and embedding (right)

In Section 14.8 we showed that every flat torus C/L is similar tothe torus spanned by τ ∈ C and 1 ∈ C, where τ is in the standardfundamental domain

D = {z = x+ iy ∈ C : |x| ≤ 12, y > 0, |z| ≥ 1}.

Definition 14.10.2. The parameter τ is called the conformal pa-rameter of the torus.

Corollary 14.10.3. The conformal parameter τ of a torus of rev-olution is pure imaginary:

τ = iσ2


of absolute value

σ2 = max

{c

d,d

c

}≥ 1.

Proof. The proof is immediate from the fact that the lattice isrectangular. �

14.11. θ-loops and φ-loops on tori of revolution

Consider a torus of revolution (T 2, g) generated by a Jordan curve Cin the (x, z)-plane, i.e., by a simple loop C, parametrized by a pair offunctions f(φ), g(φ), so that x = f(φ) and z = g(φ).

Definition 14.11.1. A φ-loop on the torus is a simple loop ob-tained by fixing the coordinate θ (i.e., the variable φ is changing). A θ-loop on the torus is a simple loop obtained by fixing the coordinate φ(i.e., the variable θ is changing).

Proposition 14.11.2. All φ-loops on the torus of revolution havethe same length equal to the length L of the generating curve C (seeCorollary 14.10.1).

Proof. The surface is rotationally invariant. In other words, allrotations around the z-axis are isometries. Therefore all φ-loops havethe same length. �

Proposition 14.11.3. The θ-loops on the torus of revolution havevariable length, depending on the φ-coordinate of the loop. Namely, thelength is 2πx = 2πf(φ).

Proof. The proof is immediate from the fact that the function f(φ)gives the distance r to the z-axis. �

Definition 14.11.4. We denote by λφ the (common) length of allφ-loops on a torus of revolution.

Definition 14.11.5. We denote by λθminthe least length of a θ-

loop on a torus of revolution, and by λθmax the maximal length of sucha θ-loop.

14.12. Tori generated by round circles

Let a, b > 0. We assume a > b so as to obtain tori that are em-bedded in 3-space. We consider the 2-parameter family ga,b of toriof revolution in 3-space with circular generating loop. The torus ofrevolution ga,b generated by a round circle is the locus of the equation

(r − a)2 + z2 = b2, (14.12.1)

14.13. CONFORMAL PARAMETER OF TORI OF REVOLUTION, RESIDUES179

where r =√x2 + y2. Note that the angle θ of the cylindrical coordi-

nates (r, θ, z) does not appear in the equation (14.12.1). The torus isobtained by rotating the circle

(x− a)2 + z2 = b2 (14.12.2)

around the z-axis in R3. The torus admits a parametrisation in termsof the functions4 f(φ) = a + b cosφ and g(φ) = b sinφ. Namely, wehave

x(θ, φ) = ((a+ b cosφ) cos θ, (a+ b cosφ) sin θ, b sinφ). (14.12.3)

Here the θ-loop (see Section 14.11) has length 2π(a+ b cosφ). Theshortest θ-loop is therefore of length

λθmin= 2π(a− b),

and the longest one is

λθmax = 2π(a+ b).

Meanwhile, the φ-loop has length

λφ = 2πb.

14.13. Conformal parameter of tori of revolution, residues

This section is optional.We would like to compute the conformal parameter τ of the stan-

dard tori as in (14.12.3). We first modify the parametrisation so as toobtain a generating curve parametrized by arclength:

f(ϕ) = a+ b cosϕ

b, g(ϕ) = b sin

ϕ

b, (14.13.1)

where ϕ ∈ [0, L] with L = 2πb.

Theorem 14.13.1. The corresponding flat torus is given by the

lattice L in the (θ, ψ) plane of the form L = SpanZ

(c ∂∂θ, d ∂

∂φ

)

where c = 2π and d = 2π√(a/b)2−1

. Thus the conformal parameter τ

of the flat torus satisfies τ = imax(((a/b)2 − 1)−1/2, ((a/b)2 − 1)1/2

).

Proof. By Corollary 14.10.3, replacing ϕ by ϕ(ψ) produces isother-mal coordinates (θ, ψ) for the torus generated by (14.13.1), where ψ =∫

dϕf(ϕ)

=∫

dϕa+b cos ϕ

b, and therefore the flat metric is defined by a lattice

in the (θ, ψ) plane with c = 2π and d =∫ L=2πb

0dϕ

a+b cos ϕb. Changing

the variable to to φ = ϕbwe obtain d =

∫ 2π

0dφ

(a/b)+cosφ, where a/b > 1.

4We use φ here and ϕ for the modified arclength parameter in the next section


Let α = a/b. Now the integral is the real part Re of the complex

integral d =∫ 2π

0dφ

α+cosφ=∫

dφα+Re(eiφ)

. Thus

d =

∫2dφ

2α + eiφ + e−iφ. (14.13.2)

The change of variables z = eiφ yields dφ = −idzz

and along the circle

we have d =∮ −2idz

z(2α+z+z−1)=∮ −2idz

z2+2αz+1=∮ −2idz

(z−λ1)(z−λ2) , where λ1 =

−α+√α2 − 1 and λ2 = −α−

√α2 − 1. The root λ2 is outside the unit

circle. Hence we need the residue at λ1 to apply the residue theorem.The residue at λ1 equals Resλ1 = −2i

λ1−λ2 = −2i2√α2−1

= −i√α2−1

. The inte-

gral is determined by the residue theorem in terms of the residue at thepole z = λ1. Therefore the lattice parameter d from Corollary 14.10.3can be computed from (14.13.2) as d =

(2πiResλ1

)= 2π√

(α)2−1. proving

the theorem. �

CHAPTER 15

A hyperreal view

15.1. Successive extensions N, Z, Q, R, ∗R

Our reference for true infinitesimal calculus is Keisler’s textbook[Ke74], downloadable athttp://www.math.wisc.edu/~keisler/calc.html

We start by motivating the familiar sequence of extensions of num-ber systems

N → Z → Q → Rin terms of their applications in arithmetic, algebra, and geometry.Each successive extension is introduced for the purpose of solving prob-lems, rather than enlarging the number system for its own sake. Thus,the extension Q ⊆ R enables one to express the length of the diagonalof the unit square and the area of the unit disc in our number system.

The familiar continuum R is an Archimedean continuum, in thesense that it satisfies the following Archimedean property.

Definition 15.1.1. An ordered field extending N is said to satisfythe Archimedean property if

(∀ǫ > 0)(∃n ∈ N) [nǫ > 1] .

We will provisionally denote the real continuum A where “A” standsfor Archimedean. Thus we obtain a chain of extensions

N → Z → Q → A,

as above. In each case one needs an enhanced ordered1 number systemto solve an ever broader range of problems from algebra or geometry.

The next stage is the extension

N → Z → Q → A → B,

where B is a Bernoullian continuum containing infinitesimals, definedas follows.

Definition 15.1.2. A Bernoullian extension of R is any properextension which is an ordered field.

1sadur

181

http://www.math.wisc.edu/~keisler/calc.html

182 15. A HYPERREAL VIEW

Any Bernoullian extension allows us to define infinitesimals and dointeresting things with those. But things become really interesting ifwe assume the Transfer Principle (Section 15.4), and work in a truehyperreal field, defined as in Definition 15.3.4 below. We will providesome motivating comments for the transfer principle in Section 15.3.

15.2. Motivating discussion for infinitesimals

Infinitesimals can be motivated from three different angles: geo-metric, algebraic, and arithmetic/analytic.

θ

Figure 15.2.1. Horn angle θ is smaller than every rec-tilinear angle

(1) Geometric (horn angles): Some students have expressedthe sentiment that they did not understand infinitesimals untilthey heard a geometric explanation of them in terms of whatwas classically known as horn angles. A horn angle is thecrivice between a circle and its tangent line at the point oftangency. If one thinks of this crevice as a quantity, it is easy toconvince oneself that it should be smaller than every rectilinearangle (see Figure 15.2.1). This is because a sufficiently smallarc of the circle will be contained in the convex region cutout by the rectilinear angle no matter how small. When onerenders this in terms of analysis and arithmetic, one gets apositive quantity smaller than every positive real number. Wecite this example merely as intuitive motivation (our actualconstruction is different).

15.3. INTRODUCTION TO THE TRANSFER PRINCIPLE 183

(2) Algebraic (passage from ring to field): The idea is torepresent an infinitesimal by a sequence tending to zero. Onecan get something in this direction without reliance on anyform of the axiom of choice. Namely, take the ring S of allsequences of real numbers, with arithmetic operations definedterm-by-term. Now quotient the ring S by the equivalencerelation that declares two sequences to be equivalent if theydiffer only on a finite set of indices. The resulting object S/Kis a proper ring extension of R, where R is embedded by meansof the constant sequences. However, this object is not a field.For example, it has zero divisors. But quotienting it furtherin such a way as to get a field, by extending the kernel K to amaximal ideal K ′, produces a field S/K ′, namely a hyperrealfield.

(3) Analytic/arithmetic: One can mimick the construction ofthe reals out of the rationals as the set of equivalence classesof Cauchy sequences, and construct the hyperreals as equiva-lence classes of sequences of real numbers under an appropriateequivalence relation.

15.3. Introduction to the transfer principle

The transfer principle is a type of theorem that, depending on thecontext, asserts that rules, laws or procedures valid for a certain num-ber system, still apply (i.e., are “transfered”) to an extended numbersystem.

Example 15.3.1. The familiar extensionQ ⊆ R preserves the prop-erty of being an ordered field.

Example 15.3.2. To give a negative example, the extension R ⊆R∪{±∞} of the real numbers to the so-called extended reals does notpreserve the property of being an ordered field.

The hyperreal extension R ⊆ ∗R (defined below) preserves all first-order properties (i.e., properties involving quantification over elementsbut not over sets.

Example 15.3.3. The formula sin2 x+cos2 x = 1, true over R for allreal x, remains valid over ∗R for all hyperreal x, including infinitesimaland infinite values of x ∈ ∗R.

Thus the transfer principle for the extension R ⊆ ∗R is a theo-rem asserting that any statement true over R is similarly true over ∗R,


and vice versa. Historically, the transfer principle has its roots in theprocedures involving Leibniz’s Law of continuity.2

We will explain the transfer principle in several stages of increasingdegree of abstraction. More details can be found in Section 15.4.

Definition 15.3.4. An ordered field B, properly including the fieldA = R of real numbers (so that A $ B) and satisfying the TransferPrinciple, is called a hyperreal field. If such an extended field B is fixedthen elements of B are called hyperreal numbers,3 while the extendedfield itself is usually denoted ∗R.

Theorem 15.3.5. Hyperreal fields exist.

For example, a hyperreal field can be constructed as the quotient ofthe ring RN of sequences of real numbers, by an appropriate maximalideal.

Definition 15.3.6. A positive infinitesimal is a positive hyperrealnumber ǫ such that

(∀n ∈ N) [nǫ < 1]

More generally, we have the following.

Definition 15.3.7. A hyperreal number ε is said to be infinitelysmall or infinitesimal if

−a < ε < a

for every positive real number a.

In particular, one has ε < 12, ε < 1

3, ε < 1

4, ε < 1

5, etc. If ε > 0

is infinitesimal then N = 1εis positive infinite, i.e., greater than every

real number.A hyperreal number that is not an infinite number are called finite.

Sometimes the term limited is used in place of finite.Keisler’s textbook exploits the technique of representing the hyper-

real line graphically by means of dots indicating the separation betweenthe finite realm and the infinite realm. One can view infinitesimals withmicroscopes as in Figure 15.6.1. One can also view infinite numberswith telescopes as in Figure 15.3.1. We have an important subset

{finite hyperreals} ⊆ ∗R

2Leibniz’s theoretical strategy in dealing with infinitesimals was analyzed ina number of detailed studies recently, which found Leibniz’ strategy to be morerobust than George Berkeley’s flawed critique thereof.

3Similar terminology is used with regard to integers and hyperintegers.

15.3. INTRODUCTION TO THE TRANSFER PRINCIPLE 185

Finite

−2 −1 0 1 2

Positive

infinite

Infinite

telescope1ǫ− 1

1/ǫ

1ǫ+ 1

Figure 15.3.1. Keisler’s telescope

which is the domain of the function called the standard part function(also known as the shadow) which rounds off each finite hyperreal tothe nearest real number (for details see Section 15.6).

Example 15.3.8. Slope calculation of y = x2 at x0. Here we usestandard part function (also called the shadow)

st : {finite hyperreals} → R

(for details concerning the shadow see Section 15.6). If a curve is de-fined by y = x2 we wish to find the slope at the point x0. To this endwe use an infinitesimal x-increment ∆x and compute the correspond-ing y-increment

∆y = (x0+∆x)2−x20 = (x0+∆x+x0)(x+∆x−x0) = (2x0+∆x)∆x.

The corresponding “average” slope is therefore

∆y

∆x= 2x0 +∆x

which is infinitely close to 2x, and we are naturally led to the definitionof the slope at x0 as the shadow of ∆y

∆x, namely st

(∆y∆x

)= 2x0.


The extension principle expresses the idea that all real objects havenatural hyperreal counterparts. We will be mainly interested in sets,functions and relations. We then have the following extension principle.

Extension principle. The order relation on the hyperreals con-tains the order relation on the reals. There exists a hyperreal numbergreater than zero but smaller than every positive real. Every set D ⊆ Rhas a natural extension ∗D ⊆ ∗R. Every real function f with domain Dhas a natural hyperreal extension ∗f with domain ∗D.4

Here the naturality of the extension alludes to the fact that suchan extension is unique, and the coherence refers to the fact that thedomain of the natural extension of a function is the natural extensionof its domain.

A positive infinitesimal is a positive hyperreal smaller than everypositive real. A negative infinitesimal is a negative hyperreal greaterthan every negative real. An arbitrary infinitesimal is either a positiveinfinitesimal, a negative infinitesimal, or zero.

Ultimately it turns out counterproductive to employ asterisks forhyperreal functions (in fact we already dropped it in equation (15.4.1)).

15.4. Transfer principle

Definition 15.4.1. The Transfer Principle asserts that every first-order statement true over R is similarly true over ∗R, and vice versa.

Here the adjective first-order alludes to the limitation on quantifi-cation to elements as opposed to sets.

Listed below are a few examples of first-order statements.

Example 15.4.2. The commutativity rule for addition x+y = y+xis valid for all hyperreal x, y by the transfer principle.

Example 15.4.3. The formula

sin2 x+ cos2 x = 1 (15.4.1)

is valid for all hyperreal x by the transfer principle.

Example 15.4.4. The statement

0 < x < y =⇒ 0 <1

y<

1

x(15.4.2)

holds for all hyperreal x, y.

4Here the noun principle (in extension principle) means that we are going toassume that there is a function ∗f : B → B which satisfies certain properties.It is a separate problem to define B which admits a coherent definition of ∗f forall f : A → A, to be solved below.

15.5. ORDERS OF MAGNITUDE 187

Example 15.4.5. The characteristic function χQ of the rationalnumbers equals 1 on rational inputs and 0 on irrational inputs. Bythe transfer principle, its natural extension ∗χQ = χ∗Q will be 1 onhyperrational numbers ∗Q and 0 on hyperirrational numbers (namely,numbers in the complement ∗R \ ∗Q).

To give additional examples of real statements to which transferapplies, note that all ordered field -statements are subject to Trans-fer. As we will see below, it is possible to extend Transfer to a muchbroader category of statements, such as those containing the functionsymbols exp or sin or those that involve infinite sequences of reals.

A hyperreal number x is finite if there exists a real number r suchthat |x| < r. A hyperreal number is called positive infinite if it isgreater than every real number, and negative infinite if it is smallerthan every real number.

15.5. Orders of magnitude

Hyperreal numbers come in three orders of magnitude: infinitesi-mal, appreciable, and infinite. A number is appreciable if it is finitebut not infinitesimal. In this section we will outline the rules for ma-nipulating hyperreal numbers.

To give a typical proof, consider the rule that if ǫ is positive in-finitesimal then 1

ǫis positive infinite. Indeed, for every positive real r

we have 0 < ǫ < r. It follows from (15.4.2) by transfer that that 1ǫis

greater than every positive real, i.e., that 1ǫis infinite.

Let ǫ, δ denote arbitrary infinitesimals. Let b, c denote arbitraryappreciable numbers. Let H,K denote arbitrary infinite numbers. Wehave the following theorem.

Theorem 15.5.1. We have the following rules for addition:

• ǫ+ δ is infinitesimal;• b+ ǫ is appreciable;• b+ c is finite (possibly infinitesimal);• H + ǫ and H + b are infinite.

We have the following rules for products.

• ǫδ and bǫ are infinitesimal;• bc is appreciable;• Hb and HK are infinite.

We have the following rules for quotients.

• ǫb, ǫH, bH

are infinitesimal;

• bcis appreciable;


• bǫ, Hǫ, Hbare infinite provided ǫ 6= 0.

We have the following rules for roots, where n is a standard naturalnumber.

• if ǫ > 0 then n√ǫ is infinitesimal;

• if b > 0 then n√b is appreciable;

• if H > 0 then n√H is infinite.

Note that the traditional topic of the so-called “indeterminate forms”can be treated without introducing any ad-hoc terminology by meansof the following remark.

Remark 15.5.2. We have no rules in certain cases, such as ǫδ, HK,Hǫ,

and H +K.

These cases correspond to what are known since Moigno5 as inde-terminate forms.

Theorem 15.5.3. Arithmetic operations on the hyperreal numbersare governed by the following rules.

(1) every hyperreal number between two infinitesimals is infinites-imal.

(2) Every hyperreal number which is between two finite hyperrealnumbers, is finite.

(3) Every hyperreal number which is greater than some positiveinfinite number, is positive infinite.

(4) Every hyperreal number which is less than some negative infi-nite number, is negative infinite.

Example 15.5.4. The difference√H + 1 −

√H − 1 (where H is

infinite) is infinitesimal. Namely,

√H + 1−

√H − 1 =

(√H + 1−

√H − 1

) (√H + 1 +

√H − 1

)(√

H + 1 +√H − 1

)

=H + 1− (H − 1)(√H + 1 +

√H − 1

)

=2√

H + 1 +√H − 1

is infinitesimal. Once we introduce limits, this example can be refor-mulated as follows: limn→∞

(√n+ 1−

√n− 1

)= 0.

5Elaborate on this historical note.

15.7. DIFFERENTIATION 189

Definition 15.5.5. Two hyperreal numbers a, b are said to be in-finitely close, written

a ≈ b,

if their difference a− b is infinitesimal.

It is convenient also to introduce the following terminology andnotation.

Definition 15.5.6. Two nonzero hyperreal numbers a, b are saidto be adequal, written

a pq b,

if either ab≈ 1 or a = b = 0.

Note that the relation sin x ≈ x for infinitesimal x is immediatefrom the continuity of sine at the origin (in fact both sides are infin-itely close to 0), whereas the relation sin x pq x is a subtler relationequivalent to the computation of the first order Taylor approximationof sine.

15.6. Standard part principle

Theorem 15.6.1 (Standard Part Principle). Every finite hyperrealnumber x is infinitely close to an appropriate real number.

Proof. The result is generally true for an arbitrary proper orderedfield extension E of R. Indeed, if x ∈ E is finite, then x induces aDedekind cut on the subfield Q ⊆ R ⊆ E via the total order of E. Thereal number corresponding to the Dedekind cut is then infinitely closeto x. �

The real number infinitely close to x is called the standard part, orshadow, denoted st(x), of x.

We will use the notation ∆x, ∆y for infinitesimals.

Remark 15.6.2. There are three consecutive stages in a typicalcalculation: (1) calculations with hyperreal numbers, (2) calculationwith standard part, (3) calculation with real numbers.

15.7. Differentiation

An infinitesimal increment ∆x can be visualized graphically bymeans of a microscope as in the Figure 15.7.1.

The slope s of a function f at a real point a is defined by setting

s = st

(f(a+∆x)− f(a)

∆x

)


r r + β r + γr + α

st

−1

−1

0

0

1

1

2

2

3

3

4

4r

Figure 15.6.1. The standard part function, st, “roundsoff” a finite hyperreal to the nearest real number. The func-tion st is here represented by a vertical projection. Keisler’s“infinitesimal microscope” is used to view an infinitesimalneighborhood of a standard real number r, where α, β,and γ represent typical infinitesimals. Courtesy ofWikipedia.

whenever the shadow exists (i.e., the ratio is finite) and is the samefor each nonzero infinitesimal ∆x. The construction is illustrated inFigure 15.7.2.

Definition 15.7.1. Let f be a real function of one real variable.The derivative of f is the new function f ′ whose value at a real x isthe slope of f at x. In symbols,

f ′(x) = st

(f(x+∆x)− f(x)

∆x

)

whenever the slope exists.

Equivalently, we can write f ′(x) ≈ f(x+∆x)−f(x)∆x

. When y = f(x)we define a new dependent variable ∆y by settting

∆y = f(x+∆x)− f(x)

called the y-increment, so we can write the derivative as st(∆y∆x

).

15.7. DIFFERENTIATION 191

a

a a+∆x

Figure 15.7.1. Infinitesimal increment ∆x under the microscope

Example 15.7.2. If f(x) = x2 we obtain the derivative of y = f(x)by the following direct calculation:

f ′(x) ≈ ∆y

∆x

=(x+∆x)2 − x2

∆x

=(x+∆x− x)(x+∆x+ x)

∆x

=∆x(2x+∆x)

∆x= 2x+∆x

≈ 2x.


a

|f(a+∆x)− f(a)|

a a+∆x

Figure 15.7.2. Defining slope of f at a

Given a function y = f(x) one defines the dependent variable ∆y =f(x+∆x)−f(x) as above. One also defines a new dependent variable dyby setting dy = f ′(x)∆x at a point where f is differentiable, and setsfor symmetry dx = ∆x. Note that we have

dy pq ∆y

as in Definition 15.5.6 whenever dy 6= 0. We then have Leibniz’s nota-tion

dy

dxfor the derivative f ′(x), and rules like the chain rule acquire an appeal-ing form.

CHAPTER 16

Global and systolic geometry

16.1. Definition of systole

The unit circle S1 ⊂ C bounds the unit disk D. A loop on asurface M is a continuous map S1 →M .

Definition 16.1.1. A loop S1 → M is called contractible if themap f extends from S1 to the disk D by means of a continuous map F :D →M .

Thus the restriction of F to S1 is f .The notions of contractible loops and simply connected spaces were

reviewed in more detail in Section 17.1.

Definition 16.1.2. A loop is called noncontractible if it is notcontractible.

Definition 16.1.3. Given a metric g onM , we will denote by sys1(g),the infimum of lengths, referred to as the “systole” of g, of a non-contractible loop β in a compact, non-simply-connected Riemannianmanifold (M, g):

sys1(g) = infβlength(β), (16.1.1)

where the infimum is over all noncontractible loops β in M . In graphtheory, a similar invariant is known as the girth [Tu47].1

It can be shown that for a compact Riemannian manifold, the in-fimum is always attained, cf. Theorem 16.13.1. A loop realizing theminimum is necessarily a simple closed geodesic.

In systolic questions about surfaces, integral-geometric identitiesplay a particularly important role. Roughly speaking, there is an inte-gral identity relating area on the one hand, and an average of energiesof a suitable family of loops, on the other. By the Cauchy-Schwarzinequality, there is an inequality relating energy and length squared,hence one obtains an inequality between area and the square of thesystole.

1The notion of systole expressed by (16.1.1) is unrelated to the systolic arraysof [Ku78].

193

194 16. GLOBAL AND SYSTOLIC GEOMETRY

Such an approach works both for the Loewner inequality (16.2.1)and Pu’s inequality (16.1.6) (biographical notes on C. Loewner andP. Pu appear in respectively). One can prove an inequality for theMobius band this way, as well [Bl61b].

Here we prove the two classical results of systolic geometry, namelyLoewner’s torus inequality as well as Pu’s inequality for the real pro-jective plane.

16.1.1. Three systolic invariants. The material in this subsec-tion is optional.

LetM be a Riemannian manifold. We define the homology 1-systole

sys1(M) (16.1.2)

by minimizing vol(α) over all nonzero homology classes. Namely, sys1(M)is the least length of a loop C representing a nontrivial homologyclass [C] in H1(M ;Z).

We also define the stable homology systole

stsys1(M) = λ1 (H1(M)/T1, ‖ ‖) , (16.1.3)

namely by minimizing the stable norm ‖ ‖ of a class of infinite order(see Definition 17.13.1 for details).

Remark 16.1.4. For the real projective plane, these two systolic in-variants are not the same. Namely, the homology systole sys1 equals theleast length of a noncontractible loop (which is also nontrivial homolog-ically), while the stable systole is infinite being defined by a minimumover an empty set.

Recall the following example from the previous section:

Example 16.1.5. For an arbitrary metric on the 2-torus T2, the1-systole and the stable 1-systole coincide by Theorem 17.5.3:

sys1(T2) = stsys1(T

2),

for every metric on T2.

Using the notion of a noncontractible loop, we can define the ho-motopy 1-systole

sys1(M) (16.1.4)

as the least length of a non-contractible loop in M .In the case of the torus, the fundamental group Z2 is abelian and

torsionfree, and therefore sys1(T2) = sys1(T

2), so that all three invari-ants coincide in this case.

16.1. DEFINITION OF SYSTOLE 195

16.1.2. Isoperimetric inequality and Pu’s inequality. Thematerial in this section is optional.

Pu’s inequality can be thought of as an “opposite” isoperimetricinequality, in the following precise sense.

The classical isoperimetric inequality in the plane is a relation be-tween two metric invariants: length L of a simple closed curve in theplane, and area A of the region bounded by the curve. Namely, everysimple closed curve in the plane satisfies the inequality

A

π≤(L

2π

)2

.

This classical isoperimetric inequality is sharp, insofar as equality isattained only by a round circle.

In the 1950’s, Charles Loewner’s student P. M. Pu [Pu52] provedthe following theorem. Let RP2 be the real projective plane endowedwith an arbitrary metric, i.e. an embedding in some Rn. Then

(L

π

)2

≤ A

2π, (16.1.5)

where A is its total area and L is the length of its shortest non-contractible loop. This isosystolic inequality, or simply systolic inequal-ity for short, is also sharp, to the extent that equality is attained onlyfor a metric of constant Gaussian curvature, namely antipodal quotientof a round sphere, cf. Section 16.14. In our systolic notation (16.1.1),Pu’s inequality takes the following form:

sys1(g)2 ≤ π

2area(g), (16.1.6)

for every metric g on RP2. See Theorem 16.16.2 for a discussion ofthe constant. The inequality is proved in http://u.math.biu.ac.il/

~katzmik/egreg826.pdf

Pu’s inequality can be generalized as follows. We will say that asurface is aspherical if it is not a 2-sphere.

Theorem 16.1.6. Every aspherical surface (M, g) satisfies the op-timal bound (16.1.6), attained precisely when, on the one hand, thesurface M is a real projective plane, and on the other, the metric g isof constant Gaussian curvature.

The extension to aspherical surfaces follows from Gromov’s inequal-ity (16.1.7) below (by comparing the numerical values of the two con-stants). Namely, every aspherical compact surface (M, g) admits ametric ball

B = Bp

(12sys1(g)

)⊆M

http://u.math.biu.ac.il/~katzmik/egreg826.pdf



of radius 12sys1(g) which satisfies [Gro83, Corollary 5.2.B]

sys1(g)2 ≤ 4

3area(B). (16.1.7)

16.1.3. Hermite and Berge-Martinet constants. The mate-rial in this subsection is optional.

Most of the material in this section has already appeared in earlierchapters.

Let b ∈ N. The Hermite constant γb is defined in one of the followingtwo equivalent ways:

(1) γb is the square of the biggest first successive minimum, cf. Defi-nition 16.17.1, among all lattices of unit covolume;

(2) γb is defined by the formula

√γb = sup

{λ1(L)

vol(Rb/L)1/b

∣∣∣∣L ⊆ (Rb, ‖ ‖)}, (16.1.8)

where the supremum is extended over all lattices L in Rb witha Euclidean norm ‖ ‖.

A lattice realizing the supremum is called a critical lattice. A crit-ical lattice may be thought of as the one realizing the densest packingin Rb when we place balls of radius 1

2λ1(L) at the points of L.

The existence of the Hermite constant, as well as the existence ofcritical lattices, are both nontrivial results [Ca71].

Theorem 14.9.1 provides the value for γ2.

Example 16.1.7. In dimensions b ≥ 3, the Hermite constants areharder to compute, but explicit values (as well as the associated crit-

ical lattices) are known for small dimensions (≤ 8), e.g. γ3 = 213 =

1.2599 . . ., while γ4 =√2 = 1.4142 . . .. Note that γn is asymptotically

linear in n, cf. (16.1.11).

A related constant γ′b is defined as follows, cf. [BeM].

Definition 16.1.8. The Berge-Martinet constant γ′b is defined bysetting

γ′b = sup{λ1(L)λ1(L

∗)∣∣L ⊆ (Rb, ‖ ‖)

}, (16.1.9)

where the supremum is extended over all lattices L in Rb.

Here L∗ is the lattice dual to L. If L is the Z-span of vectors (xi),then L∗ is the Z-span of a dual basis (yj) satisfying 〈xi, yj〉 = δij,cf. relation (16.7.1).

16.2. LOEWNER’S TORUS INEQUALITY 197

Thus, the constant γ′b is bounded above by the Hermite constant γbof (16.1.8). We have γ′1 = 1, while for b ≥ 2 we have the followinginequality:

γ′b ≤ γb ≤2

3b for all b ≥ 2. (16.1.10)

Moreover, one has the following asymptotic estimates:

b

2πe(1 + o(1)) ≤ γ′b ≤

b

πe(1 + o(1)) for b→ ∞, (16.1.11)

cf. [LaLS90, pp. 334, 337]. Note that the lower bound of (16.1.11)for the Hermite constant and the Berge-Martinet constant is noncon-structive, but see [RT90] and [ConS99].

Definition 16.1.9. A lattice L realizing the supremum in (16.1.9)or (16.1.9) is called dual-critical.

Remark 16.1.10. The constants γ′b and the dual-critical latticesin Rb are explicitly known for b ≤ 4, cf. [BeM, Proposition 2.13]. Inparticular, we have γ′1 = 1, γ′2 =

2√3.

Example 16.1.11. In dimension 3, the value of the Berge-Martinet

constant, γ′3 =√

32= 1.2247 . . ., is slightly below the Hermite con-

stant γ3 = 213 = 1.2599 . . .. It is attained by the face-centered cu-

bic lattice, which is not isodual [MilH73, p. 31], [BeM, Proposition2.13(iii)], [CoS94].

This is the end of the three subsections containing optional material.

16.2. Loewner’s torus inequality

Historically, the first lower bound for the volume of a Riemannianmanifold in terms of a systole is due to Charles Loewner. In 1949,Loewner proved the first systolic inequality, in a course on Riemanniangeometry at Syracuse University, cf. [Pu52]. Namely, he showed thefollowing result, whose proof appears in Section 19.2.

Theorem 16.2.1 (C. Loewner). Every Riemannian metric g on thetorus T2 satisfies the inequality

sys1(g)2 ≤ γ2 area(g), (16.2.1)

where γ2 = 2√3is the Hermite constant (16.1.8). A metric attaining

the optimal bound (16.2.1) is necessarily flat, and is homothetic to thequotient of C by the Eisenstein integers, i.e. lattice spanned by the cuberoots of unity, cf. Lemma 14.9.1.


The result can be reformulated in a number of ways.2 Loewner’storus inequality relates the total area, to the systole, i.e. least lengthof a noncontractible loop on the torus (T2, g):

area(g)−√32sys1(g)

2 ≥ 0. (16.2.2)

The boundary case of equality is attained if and only if the metric ishomothetic to the flat metric obtained as the quotient of R2 by thelattice formed by the Eisenstein integers.

16.3. Loewner’s inequality with remainder term

See http://u.math.biu.ac.il/~katzmik/egreg826.pdf

16.4. Global geometry of surfaces

Discussion of Local versus Global: The local behavior is by def-inition the behavior in an open neighborhood of a point. The localbehavior of a smooth curve is well understood by the implicit functiontheorem. Namely, a smooth curve in the plane or in 3-space can bethought of as the graph of a smooth function. A curve in the plane islocally the graph of a scalar function. A curve in 3-space is locally thegraph of a vector-valued function.

Example 16.4.1. The unit circle in the plane can be defined im-plicitly by

x2 + y2 = 1,

or parametrically byt 7→ (cos t, sin t).

Alternatively, it can be given locally as the graph of the function

f(x) =√1− x2.

Note that this presentation works only for points on the upper halfcir-cle. For points on the lower halfcircle we use the function

−√1− x2.

Both of these representations fail at the points (1, 0) and (−1, 0).To overcome this difficulty, we must work with y as the independent

variable, instead of x. Thus, we can parametrize a neighborhood of(1,0) by using the function

x = g(y) =√1− y2.

2Thus, in the case of the torus T2, the fundamental group is abelian. Hencethe systole can be expressed in this case as follows: sys1(T

2) = λ1(H1 (T

2;Z), ‖ ‖),

where ‖ ‖ is the stable norm.


16.5. DEFINITION OF MANIFOLD 199

Example 16.4.2. The helix given in parametric form by

(x, y, z) = (cos t, sin t, t).

It can also be defined as the graph of the vector-valued function f(z),with values in the (x, y)-plane, where

(x(z), y(z)) = f(z) = (cos z, sin z).

In this case the graph representation in fact works even globally.

Example 16.4.3. The unit sphere in 3-space can be representedlocally as the graph of the function of two variables

f(x, y) =√1− x2 − y2.

As above, concerning the local nature of the presentation necessitatesadditional functions to represent neighborhoods of points not in theopen northern hemisphere (this example is discussed in more detail inSection 16.6).

16.5. Definition of manifold

Motivated by the examples given in the previous section, we give ageneral definition as follows.

A manifold is defined as a subset of Euclidean space which is lo-cally a graph of a function, possibly vector-valued. This is the originaldefinition of Poincare who invented the notion (see Arnold [1, p. 234]).

Definition 16.5.1. By a 2-dimensional closed Riemannian mani-fold we mean a compact subset

M ⊆ Rn

such that in an open neighborhood of every point p ∈ M in Rn, thecompact subsetM can be represented as the graph of a suitable smoothvector-valued function of two variables.

Here the function has values in (n− 2)-dimensional vectors.The usual parametrisation of the graph can then be used to calcu-

late the coefficients gij of the first fundamental form, as, for example,in Theorem 16.16.2 and Example 5.10.2. The collection of all such datais then denoted by the pair (M, g), where

g = (gij)

is referred to as the metric.


Remark 16.5.2. Differential geometers like the (M, g) notation, be-cause it helps separate the topology M from the geometry g. Strictlyspeaking, the notation is redundant, since the object g already incorpo-rates all the information, including the topology. However, geometershave found it useful to use g when one wants to emphasize the geome-try, and M when one wants to emphasize the topology.

Note that, as far as the intrinsic geometry of a Riemannian mani-fold is concerned, the embedding in Rn referred to in Definition 16.5.1is irrelevant to a certain extent, all the more so since certain basicexamples, such as flat tori, are difficult to imbed in a transparent way.

16.6. Sphere as a manifold

The round 2-sphere S2 ⊆ R3 defined by the equation

x2 + y2 + z2 = 1

is a closed Riemannian manifold. Indeed, consider the function f(x, y)

defined in the unit disk x2+ y2 < 1 by setting f(x, y) =√1− x2 − y2.

Define a coordinate chart

x1(u1, u2) = (u1, u2, f(u1, u2)).

Thus, each point of the open northern hemisphere admits a neighbor-hood diffeomorphic to a ball (and hence to R2). To cover the southernhemisphere, use the chart

x2(u1, u2) = (u1, u2,−f(u1, u2)).

To cover the points on the equator, use in addition charts x3(u1, u2) =

(u1, f(u1, u2), u2), x4(u1, u2) = (u1,−f(u1, u2), u2), as well as the pair of

charts x5(u1, u2) = (f(u1, u2), u1, u2), x6(u

1, u2) = (−f(u1, u2), u1, u2).

16.7. Dual bases

Tangent space, cotangent space, and the notation for bases in thesespaces were discussed in Section 13.7.

We will work with dual bases(∂∂ui

)for vectors, and (dui) for cov-

ectors (i.e. elements of the dual space), such that

dui(

∂

∂uj

)= δij, (16.7.1)

where δij is the Kronecker delta.Recall that the metric coefficients are defined by setting

gij = 〈xi, xj〉,

16.8. JACOBIAN MATRIX 201

where x is the parametrisation of the surface. We will only work withmetrics whose first fundamental form is diagonal. We can thus writethe first fundamental form as follows:

g = g11(u1, u2)(du1)2 + g22(u

1, u2)(du2)2. (16.7.2)

Remark 16.7.1. Such data can be computed from a Euclideanembedding as usual, or it can be given apriori without an embedding,as we did in the case of the hyperlobic metric.

We will work with such data independently of any Euclidean em-bedding, as discussed in Section 10.5. For example, if the metric coef-ficients form an identity matrix, we obtain

g =(du1)2

+(du2)2, (16.7.3)

where the interior superscript denotes an index, while exterior super-script denotes the squaring operation.

16.8. Jacobian matrix

The Jacobian matrix of v = v(u) is the matrix

∂(v1, v2)

∂(u1, u2),

which is the matrix of partial derivatives. Denote by

Jacv(u) = det

(∂(v1, v2)

∂(u1, u2)

).

It is shown in advanced calculus that for any function f(v) in a do-main D, one has

∫

D

f(v)dv1dv2 =

∫

D

g(u)Jacv(u)du1du2, (16.8.1)

where g(u) = f(v(u)).

Example 16.8.1. Let u1 = r, u2 = θ. Let v1 = x, v2 = y. Wehave x = r cos θ and y = r sin θ. One easily shows that the Jacobianis Jacv(u) = r. The area elements are related by

dv1dv2 = Jacv(u)du1du2,

or

dxdy = rdrdθ.

A similar relation holds for integrals.


16.9. Area of a surface, independence of partition

Partition3 is what allows us to perform actual calculations witharea, but the result is independent of partition (see below).

Based on the local definition of area discussed in an earlier chapter,we will now deal with the corresponding global invariant.

Definition 16.9.1. The area element dA of the surface is the ele-ment

dA :=√det(gij)du

1du2,

where det(gij) = g11g22 − g212 as usual.

Theorem 16.9.2. Define the area of (M, g) by means of the formula

area =

∫

M

dA =∑

{U}

∫

U

√det(gij)du

1du2, (16.9.1)

namely by choosing a partition {U} of M subordinate to a finite opencover as in Definition 16.5.1, performing a separate integration in eachopen set, and summing the resulting areas. Then the total area is in-dependent of the partition and choice of coordinates.

Proof. Consider a change from a coordinate chart (ui) to anothercoordinate chart, denoted (vα). In the overlap of the two domains, thecoordinates can be expressed in terms of each other, e.g. v = v(u), andwe have the 2 by 2 Jacobian matrix Jacv(u).

Denote by gαβ the metric coefficients with respect to the chart (vα).Thus, in the case of a metric induced by a Euclidean embedding definedby x = x(u) = x(u1, u2), we obtain a new parametrisation

y(v) = x(u(v)).

Then we have

gαβ =

⟨∂y

∂vα,∂y

∂vβ

⟩

=

⟨∂x

∂ui∂ui

∂vα,∂x

∂uj∂uj

∂vβ

⟩

=∂ui

∂vα∂uj

∂vβ

⟨∂x

∂ui,∂x

∂uj

⟩

= gij∂ui

∂vα∂uj

∂vβ.

3ritzuf or chaluka?

16.10. CONFORMAL EQUIVALENCE 203

The right hand side is a product of three square matrices:

∂ui

∂vαgij

∂uj

∂vβ.

The matrices on the left and on the right are both Jacobian matrices.Since determinant is multiplicative, we obtain

det(gαβ) = det(gij)det

(∂(u1, u2)

∂(v1, v2)

)2

.

Hence using equation (16.8.1), we can write the area element as

dA = det12 (gαβ)dv

1dv2

= det12 (gαβ) Jacv(u)du

1du2

= det12 (gij) det

(∂(u1, u2)

∂(v1, v2)

)Jacv(u)du

1du2

= det12 (gij) du

1du2

since inverse maps have reciprocal Jacobians by chain rule. Thus theintegrand is unchanged and the area element is well defined. �

16.10. Conformal equivalence

Definition 16.10.1. Two metrics, g = gijduiduj and h = hijdu

iduj,onM are called conformally equivalent , or conformal for short, if thereexists a function f = f(u1, u2) > 0 such that

g = f 2h,

in other words,

gij = f 2hij ∀i, j. (16.10.1)

Definition 16.10.2. The function f above is called the conformalfactor (note that sometimes it is more convenient to refer, instead, tothe function λ = f 2 as the conformal factor).

Theorem 16.10.3. Note that the length of every vector at a givenpoint (u1, u2) is multiplied precisely by f(u1, u2).

Proof. More speficially, a vector v = vi ∂∂ui

which is a unit vectorfor the metric h, is “stretched” by a factor of f , i.e. its length with


respect to g equals f . Indeed, the new length of v is

√g(v, v) = g

(vi

∂

∂ui, vj

∂

∂uj

) 12

=

(g

(∂

∂ui,∂

∂uj

)vivj

) 12

=√gijvivj

=√f 2hijvivj

= f√hijvivj

= f,


Definition 16.10.4. An equivalence class of metrics on M con-formal to each other is called a conformal structure on M (mivnehconformi).

16.11. Geodesic equation

The material in this section has already been dealt with in an earlierchapter.

Perhaps the simplest possible definition of a geodesic β on a surfacein 3-space is in terms of the orthogonality of its second derivative β′′ tothe surface. The nonlinear second order ordinary differential equationdefining a geodesic is, of course, the “true” if complicated definition.We will now prove the equivalence of the two definitions. Consider aplane curve

Rs

−→α

R2

(u1,u2)

where α = (α1(s), α2(s)). Let x : R2 → R3 be a regular parametrisationof a surface in 3-space. Then the composition

Rs−→α

R2

(u1,u2)−→x

R3

yields a curve

β = x ◦ α.Definition 16.11.1. A curve β = x◦α is a geodesic on the surface x

if one of the following two equivalent conditions is satisfied:

(a) we have for each k = 1, 2,

(αk)′′

+ Γkij(αi)

′

(αj)′

= 0 where′

=d

ds, (16.11.1)

16.12. CLOSED GEODESIC 205

meaning that

(∀k) d2αk

ds2+ Γkij

dαi

ds

dαj

ds= 0;

(b) the vector β′′ is perpendicular to the surface and one has


′n. (16.11.2)

To prove the equivalence, we write β = x ◦ α, then β′ = xiαi′ by

chain rule. Furthermore,

β′′ =d

ds(xi ◦ α)αi′ + xiα

i′′ = xijαj ′αi

′+ xkα

k ′′.

Since xij = Γkijxk + Lijn holds, we have

β′′ − Lijαi′αj

′n = xk

(αk

′′+ Γkijα

i′αj′).

16.12. Closed geodesic

Definition 16.12.1. A closed geodesic in a Riemannian 2-manifoldMis defined equivalently as

(1) a periodic curve β : R → M satisfying the geodesic equa-

tion αk′′+ Γkijα

i′αj′= 0 in every chart x : R2 →M , where, as

usual, β = x◦α and α(s) = (α1(s), α2(s)) where s is arclength.Namely, there exists a period T > 0 such that β(s+T ) = β(s)for all s.

(2) A unit speed map from a circle R/LT → M satisfying thegeodesic equation at each point, where LT = TZ ⊆ R is therank one lattice generated by T > 0.

Definition 16.12.2. The length L(β) of a path β : [a, b] → M iscalculated using the formula

L(β) =

∫ b

a

‖β′(t)‖dt,

where ‖v‖ =√gijvivj whenever v = vixi. The energy is defined

by E(β) =∫ ba‖β′(t)‖2dt.

A closed geodesic as in Definition 16.12.1, item 2 has length T .

Remark 16.12.3. The geodesic equation (16.11.1) is the Euler-Lagrange equation of the first variation of arc length. Therefore when apath minimizes arc length among all neighboring paths connecting twofixed points, it must be a geodesic. A corresponding statement is validfor closed loops, cf. proof of Theorem 16.13.1. See also Section 16.1.


In Sections 16.14 and 16.18 we will give a complete description of thegeodesics for the constant curvature sphere, as well as for flat tori.

16.13. Existence of closed geodesic

Theorem 16.13.1. Every free homotopy class of loops in a closedmanifold contains a closed geodesic.

Proof. We sketch a proof for the benefit of a curious reader, whocan also check that the construction is independent of the choices in-volved. The relevant topological notions are defined in Section 17.1and [Hat02]. A free homotopy class α of a manifold M correspondsto a conjugacy class gα ⊂ π1(M). Pick an element g ∈ gα. Thus g actson the universal cover M of M . Let fg : M → R be the displacementfunction of g, i.e.

fg(x) = d(x, g.x).

Let x0 ∈M be a minimum of fg. A first variation argument shows thatany length-minimizing path between x0 and g.x0 descends to a closedgeodesic in M representing α, cf. [Car92, Ch93, GaHL04]. �

16.14. Surfaces of constant curvature

By the uniformisation theorem 13.12.2, all surfaces fall into threetypes, according to whether they are conformally equivalent to metricsthat are:

(1) flat (i.e. have zero Gaussian curvature K ≡ 0);(2) spherical (K ≡ +1);(3) hyperbolic (K ≡ −1).

For closed surfaces, the sign of the Gaussian curvature K is that ofits Euler characteristic, cf. formula (17.9.1).

Theorem 16.14.1 (Constant positive curvature). There are onlytwo compact surfaces, up to isometry, of constant Gaussian curva-ture K = +1. They are the round sphere S2 of Example 16.6; andthe real projective plane, denoted RP2.

16.15. Real projective plane

Intuitively, one thinks of the real projective plane as the quotientsurface obtained if one starts with the northern hemisphere of the 2-sphere, and “glues” together pairs of opposite points of the equatorialcircle (the boundary of the hemisphere).

More formally, the real projective plane can be defined as follows.Let

m : S2 → S2

16.16. SIMPLE LOOPS FOR SURFACES OF POSITIVE CURVATURE 207

denote the antipodal map of the sphere, i.e. the restriction of the map

v 7→ −vin R3. Then m is an involution. In other words, if we consider theaction of the group Z2 = {e,m} on the sphere, each orbit of the Z2

action on S2 consists of a pair of antipodal points

{±p} ⊆ S2. (16.15.1)

Definition 16.15.1. On the set-theoretic level, the real projectiveplane RP2 is the set of orbits of type (16.15.1), i.e. the quotient of S2

by the Z2 action.

Denote by Q : S2 → RP2 the quotient map. The smooth structureand metric on RP2 are induced from S2 in the following sense. Let x :R2 → S2 be a chart on S2 not containing any pair of antipodal points.Let gij be the metric coefficients with respect to this chart.

Let y = m◦x = −x denote the “opposite” chart, and denote by hijits metric coefficients. Then

hij = 〈∂y

∂ui,∂y

∂uj〉 = 〈− ∂x

∂ui,− ∂x

∂uj〉 = 〈 ∂x

∂ui,∂x

∂uj〉 = gij. (16.15.2)

Thus the opposite chart defines the identical metric coefficients. Thecomposition Q ◦ x is a chart on RP2, and the same functions gij formthe metric coefficients for RP2 relative to this chart.

We can summarize the preceeding discussion by means of the fol-lowing definition.

Definition 16.15.2. The real projective plane RP2 is defined inthe following two equivalent ways:

(1) the quotient of the round sphere S2 by (the restriction to S2

of) the antipodal map v 7→ −v in R3. In other words, a typicalpoint of RP2 can be thought of as a pair of opposite points ofthe round sphere.

(2) the northern hemisphere of S2, with opposite points of theequator identified.

The smooth structure of RP2 is induced from the round sphere.Since the antipodal map preserves the metric coefficients by the cal-culation (16.15.2), the metric structure of constant Gaussian curva-ture K = +1 descends to RP2, as well.

16.16. Simple loops for surfaces of positive curvature

Definition 16.16.1. A loop α : S1 → X of a space X is calledsimple if the map α is one-to-one, cf. Definition 17.1.1.


Theorem 16.16.2. The basic properties of the geodesics on surfacesof constant positive curvature as as follows:

(1) all geodesics are closed;(2) the simple closed geodesics on S2 have length 2π and are de-

fined by the great circles;(3) the simple closed geodesics on RP2 have length π;(4) the simple closed geodesics of RP2 are parametrized by half-

great circles on the sphere.

Proof. We calculate the length of the equator of S2. Here thesphere ρ = 1 is parametrized by

x(θ, ϕ) = (sinϕ cos θ, sinϕ sin θ, cosϕ)

in spherical coordinates (θ, ϕ). The equator is the curve x◦α where α(s) =(s, π

2) with s ∈ [0, 2π]. Thus α1(s) = θ(s) = s. Recall that the metric

coefficients are given by g11(θ, ϕ) = sin2 ϕ, while g22 = 1 and g12 = 0.Thus

‖β′(s)‖ =√gij(s, π

2

)αi′αj ′ =

√(sin π

2

) (dθds

)2= 1.

Thus the length of β is∫ 2π

0

‖β′(s)‖ds =∫ 2π

0

1ds = 2π.

A geodesic on RP2 is twice as short as on S2, since the antipodal pointsare identified, and therefore the geodesic “closes up” sooner than (i.e.twice as fast as) on the sphere. For example, a longitude of S2 isnot a closed curve, but it descends to a closed curve on RP2, sinceits endpoints (north and south poles) are antipodal, and are thereforeidentified with each other. �

16.17. Successive minima

The material in this section has already been dealt with in an earlierchapter.

Let B be a finite-dimensional Banach space, i.e. a vector space to-gether with a norm ‖ ‖. Let L ⊆ (B, ‖ ‖) be a lattice of maximal rank,i.e. satisfying rank(L) = dim(B). We define the notion of successiveminima of L as follows, cf. [GruL87, p. 58].

Definition 16.17.1. For each k = 1, 2, . . . , rank(L), define the k-thsuccessive minimum λk of the lattice L by

λk(L, ‖ ‖) = inf

{λ ∈ R

∣∣∣∣∃ lin. indep. v1, . . . , vk ∈ Lwith ‖vi‖ ≤ λ for all i

}. (16.17.1)

16.19. HYPERBOLIC SURFACES 209

Thus the first successive minimum, λ1(L, ‖ ‖) is the least length ofa nonzero vector in L.

16.18. Flat surfaces

A metric is called flat if its Gaussian curvature K vanishes at everypoint.

Theorem 16.18.1. A closed surface of constant Gaussian curva-ture K = 0 is topologically either a torus T2 or a Klein bottle.

Let us give a precise description in the former case.

Example 16.18.2 (Flat tori). Every flat torus is isometric to aquotient T2 = R2/L where L is a lattice, cf. [Lo71, Theorem 38.2].In other words, a point of the torus is a coset of the additive ac-tion of the lattice in R2. The smooth structure is inherited from R2.Meanwhile, the additive action of the lattice is isometric. Indeed, wehave dist(p, q) = ‖q − p‖, while for any ℓ ∈ L, we have

dist(p+ ℓ, q + ℓ) = ‖q + ℓ− (p+ ℓ)‖ = ‖q − p‖ = dist(p, q).

Therefore the flat metric on R2 descends to T2.

Note that locally, all flat tori are indistinguishable from the flatplane itself. However, their global geometry depends on the metric in-variants of the lattice, e.g. its successive minima, cf. Definition 16.17.1.Thus, we have the following.

Theorem 16.18.3. The least length of a nontrivial closed geodesicon a flat torus T2 = R2/L equals the first successive minimum λ1(L).

Proof. The geodesics on the torus are the projections of straightlines in R2. In order for a straight line to close up, it must pass througha pair of points x and x+ℓ where ℓ ∈ L. The length of the correspondingclosed geodesic on T2 is precisely ‖ℓ‖, where ‖ ‖ is the Euclidean norm.By choosing a shortest element in the lattice, we obtain a shortestclosed geodesic on the corresponding torus. �

16.19. Hyperbolic surfaces

Most closed surfaces admit neither flat metrics nor metrics of pos-itive curvature, but rather hyperbolic metrics. A hyperbolic surface isa surface equipped with a metric of constant Gaussian curvature K =−1. This case is far richer than the other two.


Example 16.19.1. The pseudosphere (so called because its Gauss-ian curvature is constant, and equals −1) is the surface of revolution

(f(ϕ) cos θ, f(ϕ) sin θ, g(ϕ))

in R3 defined by the functions f(φ) = eφ and

g(φ) =

∫ φ

0

(1− e2ψ

)1/2dψ,

where φ ranges through the interval −∞ < φ ≤ 0. The usual formu-

las g11 = f 2 as well as g22 =(drdφ

)2+(dgdφ

)2yield in our case g11 = e2φ,

while

g22 =(eφ)2 +

(√1− e2φ

)2

= e2φ + 1− e2φ

= 1.

Thus (gij) =

(e2φ 00 1

). The pseudosphere has constant Gaussian

curvature −1, but it is not a closed surface (as it is unbounded in R3).

16.20. Hyperbolic plane

This was already discussed in Section 12.3. The metric

gH2 =1

y2(dx2 + dy2) (16.20.1)

in the upper half-plane

H2 = {(x, y) | y > 0}is called the hyperbolic metric of the upper half plane.

Theorem 16.20.1. The metric (16.20.1) has constant Gaussiancurvature K = −1.

Proof. By Theorem 12.2.2, we have

K = −∆LB ln f = ∆LB ln y = y2(− 1

y2

)= −1,

as required. �

In coordinates (u1, u2), we can write it, a bit awkwardly, as

gH2 =1

(u2)2

((du1)2

+(du2)2)

.

The Riemannian manifold (H2, gH2) is referred to as the Poincare upperhalf-plane. Its significance resides in the following theorem.

16.20. HYPERBOLIC PLANE 211

Theorem 16.20.2. Every closed hyperbolic surface M is isomet-ric to the quotient of the Poincare upper half-plane by the action of asuitable group Γ:

M = H2/Γ.

Here the nonabelian group Γ is a discrete subgroup Γ ⊆ PSL(2,R),

where a matrix A =

(a bc d

)acts on H2 = {z ∈ C|ℑ(z) > 0} by

z 7→ az + b

cz + d,

called fractional linear transformations, of Mobius transformations. Allsuch transformations are isometries of the hyperbolic metric. The fol-lowing theorem is proved, for example, in [Kato92].

Theorem 16.20.3. Every geodesic in the Poincare upper half-planeis either a vertical ray, or a semicircle perpendicular to the x-axis.

The foundational significance of this model in the context of theparallel postulate of Euclid has been discussed by numerous authors.

Example 16.20.4. The length of a vertical interval joining i to cican be calculated as follows. Recall that the conformal factor is f(x, y) =1y. The length is therefore given by

∣∣∣∣∫ c

1

1

ydy

∣∣∣∣ = | ln c|.

Here the substitution y = es gives an arclength parametrisation.

CHAPTER 17

Elements of the topology of surfaces

17.1. Loops, simply connected spaces

We would like to provide a self-contained explanation of the topo-logical ingredient which is necessary so as to understand Loewner’storus inequality, i.e. essentially the notion of a noncontractible loopand the fundamental group of a topological space X. See [Hat02,Chapter 1] for a more detailed account.

Definition 17.1.1. A loop in X can be defined in one of two equiv-alent ways:

(1) a continuous map β : [a, b] → X satisfying β(a) = β(b);(2) a continuous map λ : S1 → X from the circle S1 to X.

Lemma 17.1.2. The two definitions of a loop are equivalent.

Proof. Consider the unique increasing linear function

f : [a, b] → [0, 2π]

which is one-to-one and onto. Thus, f(t) = 2π(t−a)b−a . Given a map

λ(eis) : S1 → X,

we associate to it a map β(t) = λ(eif(t)

), and vice versa. �

Definition 17.1.3. A loop S1 → X is said to be contractible ifthe map of the circle can be extended to a continuous map of the unitdisk D → X, where S1 = ∂D.

Definition 17.1.4. A space X is called simply connected if everyloop in X is contractible.

Theorem 17.1.5. The sphere Sn ⊆ Rn+1 which is the solution setof x20+ . . .+x

2n = 1 is simply connected for n ≥ 2. The circle S1 is not

simply connected.

17.2. Orientation on loops and surfaces

Let S1 ⊆ C be the unit circle. The choice of an orientation on thecircle is an arrow pointing clockwise or counterclockwise. The standard

213

214 17. ELEMENTS OF THE TOPOLOGY OF SURFACES

choice is to consider S1 as an oriented manifold with orientation chosencounterclockwise.

If a surface is embedded in 3-space, one can choose a continuousunit normal vector n at every point. Then an orientation is defined bythe right hand rule with respect to n thought of as the thumb (agudal).

17.3. Cycles and boundaries

The singular homology groups with integer coefficients, Hk(M ;Z)for k = 0, 1, . . . ofM are abelian groups which are homotopy invariantsof M . Developing the singular homology theory is time-consuming.The case that we will be primarily interested in as far as these notesare concerned, is that of the 1-dimensional homology group:

H1(M ;Z).

In this case, the homology groups can be characterized easily withoutthe general machinery of singular simplices.

Let S1 ⊆ C be the unit circle, which we think of as a 1-dimensionalmanifold with an orientation given by the counterclockwise direction.

Definition 17.3.1. A 1-cycle α on a manifold M is an integerlinear combination

α =∑

i

nifi

where ni ∈ Z is called the multiplicity (ribui), while each

fi : S1 →M

is a loop given by a smooth map from the circle to M , and each loopis endowed with the orientation coming from S1.

Definition 17.3.2. The space of 1-cycles on M is denoted

Z1(M ;Z).

Let (Σg, ∂Σg) be a surface with boundary ∂Σg, where the genus gis irrelevant for the moment and is only added so as to avoid confusionwith the summation symbol

∑.

The boundary ∂Σg is a disjoint union of circles. Now assume thesurface Σg is oriented.

Proposition 17.3.3. The orientation of the surface induces anorientation on each boundary circle.

Thus we obtain an orientation-preserving identification of each bound-ary component with the standard unit circle S1 ⊆ C (with its counter-clockwise orientation).

17.4. FIRST SINGULAR HOMOLOGY GROUP 215

Given a map Σg → M , its restriction to the boundary thereforeproduces a 1-cycle

∂Σg ∈ Z1(M ;Z).

Definition 17.3.4. The space

B1(M ;Z) ⊆ Z1(M ;Z)

of 1-boundaries in M is the space of all cycles∑

i

nifi ∈ Z1(M ;Z)

such that there exists a map of an oriented surface Σg →M (for some g)satisfying

∂Σg =∑

i

nifi.

Example 17.3.5. Consider the cylinder

x2 + y2 = 1, 0 ≤ z ≤ 1

of unit height. The two boundary components correspond to the twocircles: the “bottom” circle Cbottom defined by z = 0, and the “top”circle Ctop defined by and z = 1. Consider the orientation on thecylinder defined by the outward pointing normal vector. It induces thecounterclockwise orientation on Cbottom, and a clockwise orientationon Ctop.

Now let C0 and C1 be the same circles with the following choice oforientation: we choose a standard counterclockwise parametrisation onboth circles, i.e., parametrize them by means of (cos θ, sin θ). Then theboundary of the cylinder is the difference of the two circles: C0 − C1,or C1 − C0, depending on the choice of orientation.

Example 17.3.6. Cutting up a circle of genus 2 into two once-holedtori shows that the separating curve is a 1-boundary.

Theorem 17.3.7. On a closed orientable surface, a separating curveis a boundary, while a non-separating loop is never a boundary.

17.4. First singular homology group

Definition 17.4.1. The 1-dimensional homology group of M withinteger coefficients is the quotient group

H1(M ;Z) = Z1(M ;Z)/B1(M ;Z).

Definition 17.4.2. Given a cycle C ∈ Z1(M ;Z), its homologyclass will be denoted [C] ∈ H1(M ;Z).


Example 17.4.3. A non-separating loop on a closed surface repre-sents a non-trivial homology class of the surface.

Theorem 17.4.4. The 1-dimensional homology group H1(M ;Z) isthe abelianisation of the fundamental group π1(M):

H1(M ;Z) = (π1M)ab .

Note that a significant difference between the fundamental groupand the first homology group is the following. While only based loopsparticipate in the definition of the fundamental group, the definitionof H1(M ;Z) involves free (not based) loops.

Example 17.4.5. The fundamental groups of the real projectiveplane RP2 and the 2-torus T2 are already abelian. Therefore one ob-tains

H1(RP2;Z) = Z/2Z,and

H1(T2;Z) = Z2.

Example 17.4.6. The fundamental group of an orientable closedsurface Σg of genus g is known to be a group on 2g generators with asingle relation which is a product of g commutators. Therefore one has

H1(Σg;Z) = Z2g.

17.5. Stable norm in 1-dimensional homology

Assume the manifoldM has a Riemannian metric. Given a smoothloop f : S1 → M , we can measure its volume (length) with respect tothe metric of M . We will denote this length by

vol(f)

with a view to higher-dimensional generalisation.

Definition 17.5.1. The volume (length) of a 1-cycle C =∑

i nifiis defined as

vol(C) =∑

i

|ni| vol(fi).

Definition 17.5.2. Let α ∈ H1(M ;Z) be a 1-dimensional homol-ogy class. We define the volume of α as the infimum of volumes ofrepresentative 1-cycles:

vol(α) = inf {vol(C) | C ∈ α} ,where the infimum is over all cycles C =

∑i nifi representing the

class α ∈ H1(M ;Z).

17.6. THE DEGREE OF A MAP 217

The following phenomenon occurs for orientable surfaces.

Theorem 17.5.3. LetM be an orientable surface, i.e. 2-dimensionalmanifold. Let α ∈ H1(M ;Z). For all j ∈ N, we have

vol(jα) = j vol(α), (17.5.1)

where jα denotes the class α + α + . . .+ α, with j summands.

Proof. To fix ideas, let j = 2. By Lemma 17.5.4 below, a mini-mizing loop C representing a multiple class 2α will necessarily intersectitself in a suitable point p. Then the 1-cycle represented by C can bedecomposed into the sum of two 1-cycles (where each can be thoughtof as a loop based at p). The shorter of the two will then give a min-imizing loop in the class α which proves the identity (17.5.1) in thiscase. The general case follows similarly. �

Lemma 17.5.4. A loop going around a cylinder twice necessarilyhas a point of self-intersection.

Proof. We think of the loop as the graph of a 4π-periodic func-tion f(t) (or alternatively a function on [0, 4π] with equal values at theendpoints). Consider the difference g(t) = f(t) − f(t + 2π). Then gtakes both positive and negative values. By the intermediate functiontheorem, the function g must have a zero t0. Then f(t0) = f(t0 + 2π)hence t0 is a point of self-intersection of the loop. �

17.6. The degree of a map

An example of a degree d map is most easily produced in the caseof a circle. A self-map of a circle given by

eiθ 7→ eidθ

has degree d.We will discuss the degree in the context of surfaces only.

Definition 17.6.1. The degree

df

of a map f between closed surfaces is the algebraic number of pointsin the inverse image of a generic point of the target surface.

We can use the 2-dimensional homology groups defined elsewherein these notes, so as to calculate the degree as follows. Recall that

H2(M ;Z) = Z,

where the generator is represented by the identity self-map of the sur-face.


Theorem 17.6.2. A map

f :M1 →M2

induces a homomorphism

f∗ : H2(M1;Z) → H2(M2;Z),

corresponding to multiplication by the degree df once the groups areidentified with Z:

Z → Z, n 7→ df n.

17.7. Degree of normal map of an embedded surface

Theorem 17.7.1. Let Σ ⊆ R3 be an embedded surface. Let p be itsgenus. Consider the normal map

fn : Σ → S2

defined by sending each point x ∈ Σ to the normal vector n = nx at x.Then the degree of the normal map is precisely 1− p.

Example 17.7.2. For the unit sphere, the normal map is the iden-tity map. The genus is 0, while the degree of the normal map is 1−p =1.

Example 17.7.3. For the torus, the normal map is harder to visu-alize. The genus is 1, while the degree of the normal map is 0.

Example 17.7.4. For a genus 2 surface embedded in R3, the degreeof the normal map is 1 − 2 = −1. This means that if the surface isoriented by the outward-pointing normal vector, the normal map isorientation-reversing.

17.8. Euler characteristic of an orientable surface

The Euler characteristic χ(M) is even for closed orientable surfaces,and the integer p = p(M) ≥ 0 defined by

χ(M) = 2− 2p

is called the genus of M .

Example 17.8.1. We have p(S2) = 0, while p(T2) = 1.

In general, the genus can be understood intuitively as the number of“holes”, i.e. “handles”, in a familiar 3-dimensional picture of a pretzel.We see from formula (17.9.1) that the only compact orientable surfaceadmitting flat metrics is the 2-torus. See [Ar83] for a friendly topo-logical introduction to surfaces, and [Hat02] for a general definition ofthe Euler characteristic.

17.10. CHANGE OF METRIC EXPLOITING GAUSSIAN CURVATURE 219

17.9. Gauss–Bonnet theorem

Every embedded closed surface in 3-space admits a continuous choiceof a unit normal vector n = nx at every point x. Note that no suchchoice is possible for an embedding of the Mobius band, see [Ar83] formore details on orientability and embeddings.

Closed embedded surfaces in R3 are called orientable.

Remark 17.9.1. The integrals of type∫

M

will be understood in the sense of Theorem 16.9.2, namely using animplied partition subordinate to an atlas, and calculating the integralusing coordinates (u1, u2) in each chart, so that we can express themetric in terms of metric coefficients

gij = gij(u1, u2)

and similarly the Gaussian curvature

K = K(u1, u2).

Theorem 17.9.2 (Gauss–Bonnet theorem). Every closed surfaceMsatisfies the identity

∫

M

K(u1, u2)√

det(gij)du1du2 = 2πχ(M), (17.9.1)

where K is the Gaussian curvature function on M , whereas χ(M) isits Euler characteristic.

17.10. Change of metric exploiting Gaussian curvature

We will use the term pseudometric for a quadratic form (or theassociated bilinear form), possibly degenerate.

We would like to give an indication of a proof of the Gauss–Bonnettheorem. We will have to avoid discussing some technical points. Con-sider the normal map

F :M → S2, x 7→ nx.

Consider a neigborhood in M where the map F is a homeomorphism(this is not always possible, and is one of the technical points we areavoiding).

Definition 17.10.1. Let gM the metric of M , and h the standardmetric of S2.


Given a point x ∈M in such a neighborhood, we can calculate thecurvature K(x). We can then consider a new metric in the conformalclass of the metric gM , defined as follows.

Definition 17.10.2. We define a new pseudometric, denoted gM ,on M by multiplying by the conformal factor K(x) at the point x.Namely, gM is the pseudometric which at the point x is given by thequadratic form

gx = K(x)gx.

If K ≥ 0 then the length of vectors is multiplied by√K.

The key to understanding the proof of the Gauss–Bonnet theoremin the case of embedded surfaces is the following theorem.

Theorem 17.10.3. Consider the restriction of the normal map Fto a neighborhood as above. We modify the metric on the source M bythe conformal factor given by the Gaussian curvature, as above. Thenthe map

F : (M, gM ) → (S2, h)

preserves areas: the area of the neighborhood in M (with respect to themodified metric) equals to the “area” of its image on the sphere.

17.11. Gauss map

Definition 17.11.1. The Gauss map is the map

F :M → S2, p 7→ Np

defined by sending a point p of M the unit normal vector N = Np

thought of as a point of S2.

The map F sends an infinitesimal parallelogram on the surface, toan infinitesimal parallelogram on the sphere.

We may identify the tangent space to M at p and the tangentspace to S2 at F (p) ∈ S2. Then the differential of the map F is theWeingarten map

W : TpM → TF (p)S2.

The element of area KdA of the surface is mapped to the element ofarea of the sphere. In other words, we modify the element of areaby multiplying by the determinant (Jacobian) of the Weingarten map,namely the Gaussian curvature K(p). Hence the image of the areaelement KdA is precisely area 2-form h on the sphere, as discussed inthe previous section.

It remains to be checked that the map has topological degree givenby half the Euler characteristic of the surface M , proving the theo-rem in the case of embedded surfaces. Since degree is invariant under

17.12. AN IDENTITY 221

continuous deformations, the result can be checked for a particularstandard embedding of a surface of arbitrary genus in R3.

17.12. An identity

Another way of writing identity (17.5.1) is as follows:

vol(α) =1

jvol(jα).

This phenomenon is no longer true for higher-dimensional mani-folds. Namely, the volume of a homology class is no longer multiplica-tive. However, the limit as j → ∞ exists and is called the stable norm.

Definition 17.12.1. Let M be a compact manifold of arbitrarydimension. The stable norm ‖ ‖ of a class α ∈ H1(M ;Z) is the limit

‖α‖ = limj→∞

1

jvol(jα). (17.12.1)

It is obvious from the definition that one has ‖α‖ ≤ vol(α). How-ever, the inequality may be strict in general. As noted above, for2-dimensional manifolds we have ‖α‖ = vol(α).

Proposition 17.12.2. The stable norm vanishes for a class of finiteorder.

Proof. If α ∈ H1(M,Z) is a class of finite order, one has finitelymany possibilities for vol(jα) as j varies. The factor of 1

jin (17.12.1)

shows that ‖α‖ = 0. �

Similarly, if two classes differ by a class of finite order, their stablenorms coincide. Thus the stable norm passes to the quotient latticedefined below.

Definition 17.12.3. The torsion subgroup of H1(M ;Z) will bedenoted T1(M) ⊂ H1(M ;Z). The quotient lattice L1(M) is the lattice

L1(M) = H1(M ;Z)/T1(M).

Proposition 17.12.4. The lattice L1(M) is isomorphic to Zb1(M),where b1 is called the first Betti number of M .

Proof. This is a general result in the theory of finitely generatedabelian groups. �


17.13. Stable systole

Definition 17.13.1. Let M be a manifold endowed with a Rie-mannian metric, and consider the associated stable norm ‖ ‖. Thestable 1-systole of M , denoted stsys1(M), is the least norm of a 1-homology class of infinite order:

stsys1(M) = inf{‖α‖ | α ∈ H1 (M,Z) \ T1(M)

}

= λ1(L1(M), ‖ ‖

).

Example 17.13.2. For an arbitrary metric on the 2-torus T2, the1-systole and the stable 1-systole coincide by Theorem 17.5.3:

sys1(T2) = stsys1(T

2),

for every metric on T2.

17.14. Free loops, based loops, and fundamental group

One can refine the notion of simple connectivity by introducing agroup, denoted

π1(X) = π1(X, x0),

and called the fundamental group of X relative to a fixed “base”point x0 ∈ X.

Definition 17.14.1. A based loop is a loop α : [0, 1] → X satisfyingthe condition α(0) = α(1) = x0.

In terms of the second item of Definition 17.1.1, we choose a fixedpoint s0 ∈ S1. For example, we can choose s0 = ei0 = 1 for theusual unit circle S1 ⊆ C, and require that α(s0) = x0. Then thegroup π1(X) is the quotient of the space of all based loops modulothe equivalence equivalence relation defined by homotopies fixing thebasepoint, cf. Definition 17.14.2.

The equivalence class of the identity element is precisely the setof contractible loops based at x0. The equivalence relation can bedescribed as follows for a pair of loops in terms of the second item ofDefinition 17.1.1.

Definition 17.14.2. Two based loops α, β : S1 → X are equiva-lent, or homotopic, if there is a continuous map of the cylinder S1 ×[c, d] → X whose restriction to S1×{c} is α, whose restriction to S1×{d} is β, while the map is constant on the segment {s0} × [c, d] of thecylinder, i.e. the basepoint does not move during the homotopy.

17.15. FUNDAMENTAL GROUPS OF SURFACES 223

Definition 17.14.3. An equivalence class of based loops is calleda based homotopy class. Removing the basepoint restriction (as well asthe constancy condition of Definition 17.14.2), we obtain a larger classcalled a free homotopy class (of loops).

Definition 17.14.4. Composition of two loops is defined most con-veniently in terms of item 1 of Definition 17.1.1, by concatenating theirdomains.

In more detail, the product of a pair of loops, α : [−1, 0] → Xand β : [0,+1] → X, is a loop α.β : [−1, 1] → X, which coincideswith α and β in their domains of definition. The product loop α.βis continuous since α(0) = β(0) = x0. Then Theorem 17.1.5 can berefined as follows.

17.15. Fundamental groups of surfaces

Theorem 17.15.1. We have π1(S1) = Z, while π1(Sn) is the trivial

group for all n ≥ 2.

Definition 17.15.2. The 2-torus T2 is defined to be the followingCartesian product: T2 = S1 × S1, and can thus be realized as a subset

T2 = S1 × S1 ⊆ R2 × R2 = R4.

We have π1(T2) = Z2. The familiar doughnut picture realizes T2 asa subset in Euclidean space R3.

Definition 17.15.3. A 2-dimensional closed Riemannian manifold(i.e. surface) M is called orientable if it can be realized by a subsetof R3.

Theorem 17.15.4. The fundamental group of a surface of genus gis isomorphic to a group on 2g generators a1, b1, . . . , ag, bg satisfyingthe unique relation

g∏

i=1

aibia−1i b−1

i = 1.

Note that this is not the only possible presentation of the group interms of a single relation.

CHAPTER 18

Pu’s inequality

See http://u.math.biu.ac.il/~katzmik/egreg826.pdf

225


CHAPTER 19

Approach To Loewner via energy-area identity

19.1. An integral-geometric identity

Let T2 be a torus with an arbitrary metric. Let T0 = R2/L be theflat torus conformally equivalent to T2. Let ℓ0 = ℓ0(x) be a simple

closed geodesic of T0. Thus ℓ0 is the projection of a line ℓ0 ⊆ R2.Let ℓy ⊆ R2 be the line parallel to ℓ0 at distance y > 0 from ℓ0 (here

we must “choose sides”, e.g. by orienting ℓ0 and requiring ℓy to lie to

the left of ℓ0). Denote by ℓy ⊆ T0 the closed geodesic loop defined by

the image of ℓy. Let y0 > 0 be the smallest number such that ℓy0 = ℓ0,

i.e. the lines ℓy0 and ℓ0 both project to ℓ0.Note that the loops in the family {ℓy} ⊆ T2 are not necessarily

geodesics with respect to the metric of T2. On the other hand, thefamily satisfies the following identity.

Lemma 19.1.1 (An elementary integral-geometric identity). Themetric on T2 satisfies the following identity:

area(T2) =

∫ y0

0

E(ℓy)dy, (19.1.1)

where E is the energy of a loop with respect to the metric of T2, seeDefinition 16.12.2.

Proof. Denote by f 2 the conformal factor of T2 with respect tothe flat metric T0. Thus the metric on T2 can be written as

f 2(x, y)(dx2 + dy2).

By Fubini’s theorem applied to a rectangle with sides lengthT0(ℓ0)

and y0, combined with Theorem 16.10.3, we obtain

area(T2) =

∫

T0

f 2dxdy

=

∫ y0

0

(∫

ℓy

f 2(x, y)dx

)dy

=

∫ y0

0

E(ℓy)dy,

227

228 19. APPROACH TO LOEWNER VIA ENERGY-AREA IDENTITY


Remark 19.1.2. The identity (19.1.1) can be thought of as thesimplest integral-geometric identity.

19.2. Two proofs of the Loewner inequality

We give a slightly modified version of M. Gromov’s proof [Gro96],using conformal representation and the Cauchy-Schwarz inequality, ofthe Loewner inequality (16.2.1) for the 2-torus, see also [CK03]. Wepresent the following slight generalisation: there exists a pair of closedgeodesics on (T2, g), of respective lengths λ1 and λ2, such that

λ1λ2 ≤ γ2 area(g), (19.2.1)

and whose homotopy classes form a generating set for π1(T2) = Z×Z.

Proof. The proof relies on the conformal representation

φ : T0 → (T2, g),

where T0 is flat, cf. uniformisation theorem 13.12.2. Here the map φmay be chosen in such a way that (T2, g) and T0 have the same area.Let f be the conformal factor, so that

g = f 2((du1)2 + (du2)2

)

as in formula (16.10.1), where (du1)2+(du2)2 (locally) is the flat metric.Let ℓ0 be any closed geodesic in T0. Let {ℓs} be the family of

geodesics parallel to ℓ0. Parametrize the family {ℓs} by a circle S1 oflength σ, so that

σℓ0 = area(T0).

Thus T0 → S1 is a Riemannian submersion. Then

area(T2) =

∫

T0

f 2.

By Fubini’s theorem, we have area(T2) =∫S1 ds

∫ℓsf 2dt. Therefore by

the Cauchy-Schwarz inequality,

area(T2) ≥∫

S1

ds

(∫ℓsfdt)2

ℓ0=

1

ℓ0

∫

S1

ds (lengthφ(ℓs))2 .

Hence there is an s0 such that area(T2) ≥ σℓ0lengthφ(ℓs0)

2, so that

lengthφ(ℓs0) ≤ ℓ0.

This reduces the proof to the flat case. Given a lattice in C, we choosea shortest lattice vector λ1, as well as a shortest one λ2 not proportionalto λ1. The inequality now follows from Lemma 14.9.1. In the boundary

19.2. TWO PROOFS OF THE LOEWNER INEQUALITY 229

case of equality, one can exploit the equality in the Cauchy-Schwarzinequality to prove that the conformal factor must be constant. �

Alternative proof. Let ℓ0 be any simple closed geodesic in T0.Since the desired inequality (19.2.1) is scale-invariant, we can assumethat the loop has unit length:

lengthT0(ℓ0) = 1,

and, moreover, that the corresponding covering transformation of theuniversal cover C = T0 is translation by the element 1 ∈ C. Wecomplete 1 to a basis {τ, 1} for the lattice of covering transformationsof T0. Note that the rectangle defined by

{z = x+ iy ∈ C | 0 < x < 1, 0 < y < Im(τ)]}is a fundamental domain for T0, so that area(T0) = Im(τ). The maps

ℓy(x) = x+ iy, x ∈ [0, 1]

parametrize the family of geodesics parallel to ℓ0 on T0. Recall thatthe metric of the torus T is f 2(dx2 + dy2).

Lemma 19.2.1. We have the following relation between the lengthand energy of a loop:

length(ℓy)2 ≤ E(ℓy).

Proof. By the Cauchy-Schwarz inequality,

∫ 1

0

f 2(x, y)dx ≥(∫ 1

0

f(x, y)dx

)2

,


Now by Lemma 19.1.1 and Lemma 19.2.1, we have

area(T2) ≥∫ Im(τ)

0

(length(ℓy)

)2dy.

Hence there is a y0 such that

area(T2) ≥ Im(τ) length(ℓy0)2,

so that length(ℓy0) ≤ 1. This reduces the proof to the flat case. Givena lattice in C, we choose a shortest lattice vector λ1, as well as ashortest one λ2 not proportional to λ1. The inequality now followsfrom Lemma 14.9.1. �

230 19. APPROACH TO LOEWNER VIA ENERGY-AREA IDENTITY

19.3. Remark on capacities

Define a conformal invariant called the capacity of an annulus asfollows. Consider a right circular cylinder

ζκ = R/Z× [0, κ]

based on a unit circle R/Z. Its capacity C(ζκ) is defined to be itsheight, C(ζκ) = κ. Recall that every annular region in the plane is con-formally equivalent to such a cylinder, and therefore we have defined aconformal invariant of an arbitrary annular region. Every annular re-gion R satisfies the inequality area(R) ≥ C(R) sys1(R)

2. Meanwhile, ifwe cut a flat torus along a shortest loop, we obtain an annular region R

with capacity at least C(R) ≥ γ−12 =

√32. This provides an alternative

proof of the Loewner theorem. In fact, we have the following identity:

confsys1(g)2C(g) = 1, (19.3.1)

where confsys is the conformally invariant generalisation of the homol-ogy systole, while C(g) is the largest capacity of a cylinder obtainedby cutting open the underlying conformal structure on the torus.

Question 19.3.1. Is the Loewner inequality (16.2.1) satisfied byevery orientable nonsimply connected compact surface? Inspite of itselementary nature, and considerable research devoted to the area, thequestion is still open. Recently the case of genus 2 was settled as wellas genus g ≥ 20

19.4. A table of optimal systolic ratios of surfaces

Denote by SR(M) the supremum of the systolic ratios,

SR(M) = supg

sys1(g)2

area(g),

ranging over all metrics g on a surface M . The known values of theoptimal systolic ratio are tabulated in Figure 19.4.1. It is interesting tonote that the optimal ratio for the Klein bottle RP2#RP2 is achievedby a singular metric, described in the references listed in the table.

19.4. A TABLE OF OPTIMAL SYSTOLIC RATIOS OF SURFACES 231

SR(M) numerical value where to find it

M = RP2 =π

2[Pu52] ≈ 1.5707 site for 88826

infinite π1(M) <4

3[Gro83] < 1.3333 . . . (16.1.7)

M = T2 = 2√3(Loewner) ≈ 1.1547 (16.2.1)

M =RP2#RP2

=π

23/2≈ 1.1107 [Bav86, Bav06,

Sak88]

M of genus 2 > 13

(√2 + 1

)> 0.8047 ?

M of genus 3 ≥ 87√3

> 0.6598 [Cal96]

Figure 19.4.1. Values for optimal systolic ratio SR of surface M

CHAPTER 20

A primer on surfaces

In this Chapter, we collect some classical facts on Riemann surfaces.More specifically, we deal with hyperelliptic surfaces, real surfaces, andKatok’s optimal bound for the entropy of a surface.

20.1. Hyperelliptic involution

Let M be an orientable closed Riemann surface which is not asphere. By a Riemann surface, we mean a surface equipped with afixed conformal structure, cf. Definition 16.10.4, while all maps areangle-preserving.

Furthermore, we will assume that the genus is at least 2.

Definition 20.1.1. A hyperelliptic involution of a Riemann sur-face M of genus p is a holomorphic (conformal) map, J : M → M ,satisfying J2 = 1, with 2p + 2 fixed points.

Definition 20.1.2. A surface M admitting a hyperelliptic involu-tion will be called a hyperelliptic surface.

Remark 20.1.3. The involution J can be identified with the non-trivial element in the center of the (finite) automorphism group of M(cf. [FK92, p. 108]) when it exists, and then such a J is unique,cf. [Mi95, p.204] (recall that the genus is at least 2).

It is known that the quotient of M by the involution J produces aconformal branched 2-fold covering

Q :M → S2 (20.1.1)

of the sphere S2.

Definition 20.1.4. The 2p + 2 fixed points of J are called Weier-strass points. Their images in S2 under the ramified double cover Q offormula (20.1.1) will be referred to as ramification points.

A notion of a Weierstrass point exists on any Riemann surface, butwill only be used in the present text in the hyperelliptic case.

233

234 20. A PRIMER ON SURFACES

Example 20.1.5. In the case p = 2, topologically the situationcan be described as follows. A simple way of representing the figure 8contour in the (x, y) plane is by the reducible curve

(((x− 1)2 + y2)− 1)(((x+ 1)2 + y2)− 1) = 0 (20.1.2)

(or, alternatively, by the lemniscate r2 = cos 2θ in polar coordinates,i.e. the locus of the equation (x2 + y2)2 = x2 − y2).

Now think of the figure 8 curve of (20.1.2) as a subset of R3. Theboundary of its tubular neighborhood in R3 is a genus 2 surface. Rota-tion by π around the x-axis has six fixed points on the surface, namely,a pair of fixed points near each of the points −2, 0, and +2 on the x-axis. The quotient by the rotation can be seen to be homeomorphic tothe sphere.

A similar example can be repackaged in a metrically more preciseway as follows.

Example 20.1.6. We start with a round metric on RP2. Nowattach a small handle. The orientable double cover M of the resultingsurface can be thought of as the unit sphere in R3, with two littlehandles attached at north and south poles, i.e. at the two points wherethe sphere meets the z-axis. Then one can think of the hyperellipticinvolution J as the rotation of M by π around the z-axis. The sixfixed points are the six points of intersection of M with the z-axis.Furthermore, there is an orientation reversing involution τ onM , givenby the restriction toM of the antipodal map in R3. The composition τ◦J is the reflection fixing the xy-plane, in view of the following matrixidentity:

−1 0 00 −1 00 0 −1

−1 0 00 −1 00 0 1

=

1 0 00 1 00 0 −1

. (20.1.3)

Meanwhile, the induced orientation reversing involution τ0 on S2 canjust as well be thought of as the reflection in the xy-plane. This isbecause, at the level of the 2-sphere, it is “the same thing as” thecomposition τ ◦ J . Thus the fixed circle of τ0 is precisely the equator,cf. formula (20.3.3). Then one gets a quotient metric on S2 which isroughly that of the western hemisphere, with the boundary longitudefolded in two, The metric has little bulges along the z-axis at northand south poles, which are leftovers of the small handle.

20.3. OVALLESS SURFACES 235

20.2. Hyperelliptic surfaces

For a treatment of hyperelliptic surfaces, see [Mi95, p. 60-61].By [Mi95, Proposition 4.11, p. 92], the affine part of a hyperellip-tic surface M is defined by a suitable equation of the form

w2 = f(z) (20.2.1)

in C2, where f is a polynomial. On such an affine part, the map J isgiven by J(z, w) = (z,−w), while the hyperelliptic quotient map Q :M → S2 is represented by the projection onto the z-coordinate in C2.

A slight technical problem here is that the map

M → CP2, (20.2.2)

whose image is the compactification of the solution set of (20.2.1), isnot an embedding. Indeed, there is only one point at infinity, givenin homogeneous coordinates by [0 : w : 0]. This point is a singularity.A way of desingularizing it using an explicit change of coordinates atinfinity is presented in [Mi95, p. 60-61]. The resulting smooth surfaceis unique [DaS98, Theorem, p. 100].

Remark 20.2.1. To explain what happens “geometrically”, notethat there are two points on our affine surface “above infinity”. Thismeans that for a large circle |z| = r, there are two circles above it satis-fying equation (20.2.1) where f has even degree 2p+2 (for a Weierstrasspoint we would only have one circle). To see this, consider z = reia. Asthe argument a varies from 0 to 2π, the argument of f(z) will changeby (2p + 2)2π. Thus, if (reia, w(a)) represents a continuous curve onour surface, then the argument of w changes by (2p+2)π, and hence weend up where we started, and not at −w (as would be the case were thepolynomial of odd degree). Thus there are two circles on the surfaceover the circle |z| = r. We conclude that to obtain a smooth compactsurface, we will need to add two points at infinity, cf. discussion around[FK92, formula (7.4.1), p. 102].

Thus, the affine part of M , defined by equation (20.2.1), is a Rie-mann surface with a pair of punctures p1 and p2. A neighborhood ofeach pi is conformally equivalent to a punctured disk. By replacingeach punctured disk by a full one, we obtain the desired compact Rie-mann surface M . The point at infinity [0 : w : 0] ∈ CP2 is the imageof both pi under the map (20.2.2).

20.3. Ovalless surfaces

Denote by M ι the fixed point set of an involution ι of a Riemannsurface M . Let M be a hyperelliptic surface of even genus p. Let J :


M → M be the hyperelliptic involution, cf. Definition 20.1.1. Let τ :M →M be a fixed point free, antiholomorphic involution.

Lemma 20.3.1. The involution τ commutes with J and descendsto S2. The induced involution τ0 : S

2 → S2 is an inversion in a circleC0 = Q(M τ◦J). The set of ramification points is invariant under theaction of τ0 on S2.

Proof. By the uniqueness of J , cf. Remark 20.1.3, we have thecommutation relation

τ ◦ J = J ◦τ, (20.3.1)

cf. relation (20.1.3). Therefore τ descends to an involution τ0 on thesphere. There are two possibilities, namely, τ is conjugate, in the groupof fractional linear transformations, either to the map z 7→ z, or to themap z 7→ −1

z. In the latter case, τ is conjugate to the antipodal map

of S2.In the case of even genus, there is an odd number of Weierstrass

points in a hemisphere. Hence the inverse image of a great circle is aconnected loop. Thus we get an action of Z2 × Z2 on a loop, resultingin a contradiction.

In more detail, the set of the 2p + 2 ramification points on S2 iscentrally symmetric. Since there is an odd number, p + 1, of ramifi-cation points in a hemisphere, a generic great circle A ⊆ S2 has theproperty that its inverse image Q−1(A) ⊆ M is connected. Thus bothinvolutions τ and J , as well as τ ◦ J , act fixed point freely on theloop Q−1(A) ⊆M , which is impossible. Therefore τ0 must fix a point.It follows that τ0 is an inversion in a circle. �

Suppose a hyperelliptic Riemann surfaceM admits an antiholomor-phic involution τ . In the literature, the components of the fixed pointsetM τ of τ are sometimes referred to as “ovals”. When τ is fixed pointfree, we introduce the following terminology.

Definition 20.3.2. A hyperelliptic surface (M,J) of even positivegenus p > 0 is called ovalless real if one of the following equivalentconditions is satisfied:

(1) M admits an imaginary reflection, i.e. a fixed point free, an-tiholomorphic involution τ ;

(2) the affine part of M is the locus in C2 of the equation

w2 = −P (z), (20.3.2)

where P is a monic polynomial, of degree 2p + 2, with realcoefficients, no real roots, and with distinct roots.

20.4. KATOK’S ENTROPY INEQUALITY 237

Lemma 20.3.3. The two ovalless reality conditions of Definition 20.3.2are equivalent.

Proof. A related result appears in [GroH81, p. 170, Proposi-tion 6.1(2)]. To prove the implication (2) =⇒ (1), note that complexconjugation leaves the equation invariant, and therefore it also leavesinvariant the locus of (20.3.2). A fixed point must be real, but P is pos-itive hence (20.3.2) has no real solutions. There is no real solution atinfinity, either, as there are two points at infinity which are not Weier-strass points, since P is of even degree, as discussed in Remark 20.2.1.The desired imaginary reflection τ switches the two points at infinity,and, on the affine part of the Riemann surface, coincides with complexconjugation (z, w) 7→ (z, w) in C2.

To prove the implication (1) =⇒ (2), note that by Lemma 20.3.1,the induced involution τ0 on S

2 =M/ J may be thought of as complexconjugation, by choosing the fixed circle of τ0 to be the circle

R ∪ {∞} ⊆ C ∪ {∞} = S2. (20.3.3)

Since the surface is hyperelliptic, it is the smooth completion of thelocus in C2 of some equation of the form (20.3.2), cf. (20.2.1). Here Pis of degree 2p+2 with distinct roots, but otherwise to be determined.The set of roots of P is the set of (the z-coordinates of) the Weierstrasspoints. Hence the set of roots must be invariant under τ0. Thus theroots of the polynomial either come in conjugate pairs, or else are real.Therefore P has real coefficients. Furthermore, the leading cofficientof P may be absorbed into the w-coordinate by extracting a squareroot. Here we may have to rotate w by i, but at any rate the coefficientsof P remain real, and thus P can be assumed monic.

If P had a real root, there would be a ramification point fixed by τ0.But then the corresponding Weierstrass point must be fixed by τ , aswell! This contradicts the fixed point freeness of τ . Thus all roots of Pmust come in conjugate pairs. �

20.4. Katok’s entropy inequality

Let (M, g) be a closed surface with a Riemannian metric. De-note by (M, g) the universal Riemannian cover of (M, g). Choose apoint x0 ∈ M .

Definition 20.4.1. The volume entropy (or asymptotic volume) h(M, g)of a surface (M, g) is defined by setting

h(M, g) = limR→+∞

log (volg B(x0, R))

R, (20.4.1)


where volg B(x0, R) is the volume (area) of the ball of radius R centered

at x0 ∈ M .

Since M is compact, the limit in (20.4.1) exists, and does not de-pend on the point x0 ∈ M [Ma79]. This asymptotic invariant describesthe exponential growth rate of the volume in the universal cover.

Definition 20.4.2. The minimal volume entropy, MinEnt, of M isthe infimum of the volume entropy of metrics of unit volume on M , orequivalently

MinEnt(M) = infgh(M, g) vol(M, g)

12 (20.4.2)

where g runs over the space of all metrics on M . For an n-dimensionalmanifold in place of M , one defines MinEnt similarly, by replacing theexponent of vol by 1

n.

The classical result of A. Katok [Kato83] states that every metric gon a closed surfaceM with negative Euler characteristic χ(M) satisfiesthe optimal inequality

h(g)2 ≥ 2π|χ(M)|area(g)

. (20.4.3)

Inequality (20.4.3) also holds for hom ent(g) [Kato83], as well as thetopological entropy, since the volume entropy bounds from below thetopological entropy (see [Ma79]). We recall the following well-knownfact, cf. [KatH95, Proposition 9.6.6, p. 374].

Lemma 20.4.3. Let (M, g) be a closed Riemannian manifold. Then,

h(M, g) = limT→+∞

log(P ′(T ))

T(20.4.4)

where P ′(T ) is the number of homotopy classes of loops based at somefixed point x0 which can be represented by loops of length at most T .

Proof. Let x0 ∈M and choose a lift x0 ∈ M . The group

Γ := π1(M,x0)

acts on M by isometries. The orbit of x0 under Γ is denoted Γ.x0.Consider a fundamental domain ∆ for the action of Γ, containing x0.Denote by D the diameter of ∆. The cardinal of Γ.x0 ∩ B(x0, R) isbounded from above by the number of translated fundamental do-mains γ.∆, where γ ∈ Γ, contained in B(x0, R+D). It is also bounded

20.4. KATOK’S ENTROPY INEQUALITY 239

from below by the number of translated fundamental domains γ.∆contained in B(x0, R). Therefore, we have

vol(B(x0, R))

vol(M, g)≤ card (Γ.x0 ∩ B(x0, R)) ≤

vol(B(x0, R +D))

vol(M, g).

(20.4.5)Take the log of these terms and divide by R. The lower bound becomes

1

Rlog

(vol(B(R))

vol(g)

)=

=1

Rlog (vol(B(R)))− 1

Rlog (vol(g)) ,

(20.4.6)

and the upper bound becomes

1

Rlog

(vol(B(R +D))

vol(g)

)=

=R +D

R

1

R +Dlog(vol(B(R +D)))− 1

Rlog(vol(g)).

(20.4.7)

Hence both bounds tend to h(g) when R goes to infinity. Therefore,

h(g) = limR→+∞

1

Rlog (card(Γ.x0 ∩ B(x0, R))) . (20.4.8)

This yields the result since P ′(R) = card(Γ.x0 ∩ B(x0, R)). �

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analytic and differential geometry mikhail g. katz∗ - cs@biu

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